• $a_1 = |\mathcal{A}|$ edges.

• $a_2 = a_2^o + a_2^n + a_2^r$ faces, where $a_2^o$ counts the ordinary faces where, $a_2^n$ the faces that are topological disks but not ordinary, and $a_2^r$ the faces that are not topological disks;

Faces counted by $a_2^n$ contain a Fuchsian singularity or a swath neighborhood (if it contains an irregular singularity then it contains a swath neighborhood) and a multiple pole of order $d_i$ has $d_i - 1$ swathes, in particular:

$$a_2^n \le n'_1 + n_2 + \sum_{i=1}^{k} (d_i - 1).$$

The Euler characteristic of $X$ satisfies

$$\chi(X) = 2 - 2g = a_0 - a_1 + \sum_F \chi(F)$$

where the sum is over all faces $F$, and $\chi(F)$ is the Euler characteristic of $F$ (as an open set). So

$$|\mathcal{A}| = 2g - 2 + a_0 + \sum_F \chi(F).$$

Since $\mathcal{A}$ is non-empty, no face is a sphere, hence the faces homeomorphic to disks satisfy $\chi(F) = 1$ while the others $\chi(F) \le 0$. We thus get

$$|\mathcal{A}| \le 2g - 2 + a_0 + a_2^o + a_2^n.$$

Since each ordinary face is at least 3-sided we get $2|\mathcal{A}| \ge 3a_2^o$, so $a_2^o \le \frac{2}{3}|\mathcal{A}|$, whence $|\mathcal{A}| \le 2g - 2 + a_0 + \frac{2}{3}|\mathcal{A}| + a_2^n$ i.e.

$$|\mathcal{A}| \le 6g - 6 + 3(a_0 + a_2^n).$$

Finally, $a_0 + a_2^n = a_0'' + a_0' + a_2^n \le a_0'' + a_0' + n'_1 + n_2 + \sum_{i=1}^k (d_i - 1) = (a_0'' - k) + n_1 + n_2 + \sum_{i=1}^k d_i = (a_0'' - k) + n \le n$. $\square$

**Remark 4.6.** Observe in particular that Equation (4.3) implies $2g+n > 2$ ($\mathcal{A}$ is non-empty by hypothesis).

We deduce an upper bound on the maximal number of non-intersecting saddle connections. We recall that a saddle connection is a geodesic connecting two (possibly identical) singular points in finite time. We call a saddle connection *simple* if it is injective, except that the two ends may or may not coincide.

**Corollary 4.7.** Consider a finite type affine surface $(X, \nabla, S)$ where $X$ is a compact Riemann surface of genus $g$ while $S$ has $n$ singular points (counted with multiplicity).

• If $2g + n \ge 3$, any family $\mathcal{F}$ of simple saddle connections with disjoint interiors has at most $6g - 6 + 3n$ elements.

• If $2g + n \le 2$, then $(X, \nabla, S)$ does not admit any saddle connection, simple or not.

*Proof.* The case $2g + n \le 2$ is handled by Proposition 2.24, so we assume that $2g + n \ge 3$. Then $\mathcal{F}$ is an apical arc system. Without loss of generality we may also assume that $\mathcal{F}$ is finite. Then $\mathcal{F}$ is an apical arc system. If $\mathcal{F}$ is then $|\mathcal{F}| = 0$ while $6g - 6 + 3n \ge 3$ so the inequality holds. In the rest of the proof, we assume $\mathcal{F}$ non-empty.

We claim that the system $\mathcal{F}$ then satisfies the assumptions of Lemma 4.5. Otherwise, a developing map of the affine structure would map any ordinary face with less than 3 sides to a connected open subset $U$ of $\mathbb{C}$ of finite area and whose boundary is contained in the union of two straight segments. There is no such domain $U$.

Then Lemma 4.5 gives the result. $\square$

**Lemma 4.8.** If $2g + n \ge 3$ then $(X, \nabla, S)$ has at least one apex.
Proof. If $\nabla$ has a multiple pole then it correspond to an irregular singularity, which has at least one focus, so $X$ has an apex. If every pole of $\nabla$ is simple. Recall (see Section 2.4) that a simple pole is a conical singularity if and only if its residue res satisfies $\text{Re} \ res < 1$. From $2g + n \ge 3$ we get three cases:

• (a): $g \ge 2$ and at least one pole is a conical singularity;

• (b): $g = 1$ and $n \ge 1$ so at least one singular point (possibly a marked point) is a conical singularity;

• (c): $g = 0$, $n \ge 3$, the total residue is 2 so one of them has a real part of at most $\frac{2}{3}$ and determines a conical singularity.

This proves the claim.

□

## 4.2. Neighborhoods of geodesics.
Before entering in the subject, we prove a preliminary lemma that will be useful here and elsewhere. In this section we consider a Riemann surface $S$ with a meromorphic connection $\nabla$. By an affine immersion we mean a locally affine map from an open subset of an affine surface and taking values in the complement of the poles of $S$.

**Lemma 4.9.** Let $S$ be a Riemann surface with a meromorphic connection $\nabla$. Let $p \in S$ be (1) a regular point, (2) a conical singularity or (3) a focus of an irregular singularity. For any $R > 0$ and any neighborhood $V'$ of $p$, there exists a punctured neighborhood $V \subset V'$ of $p$ in cases (1) and (2), a repelling petal neighborhood of the focus in case (3), such that $\forall x \in V$ in cases (1) and (2) and $\forall x \in V \cap P$ in case (3), there is an affine immersion (possibly injective) $f$ from the slit disk

$$B(0, R) - (-R, 0] = \text{Dom}(f)$$

to $V'$, sending 1 to $x$ and such that $f(z)$ tends to $p$ as $z$ tends to 0 within $\text{Dom}(f)$

**Proof.** For a regular singularity the claim is very easy and we do not need the slit. Below we assume $p$ is not regular.

Recall that the exponential-affine plane has two models, one as a universal covering $\tilde{\mathbb{C}}^*$ of $\mathbb{C}^*$ endowed with the restriction of the canonical affine structure of $\mathbb{C}$, in which we use name the coordinate $u$. One as $\mathcal{E} = \mathbb{C}$ where we name the coordinate $w$, endowed with the connection with $\Gamma_w = 1$, or equivalently for which the map $w \mapsto u = \exp(w)$ is affine.

For a conical singularity, a neighborhood $V_0$ of $p$ is isomorphic, for some $\tau \in \mathbb{C}$ with $\text{Im} \tau > 0$ to the quotient (see Sections 2.4 and 2.6) of the subset $H$ of $\mathcal{E}$ of equation $\text{Im}(w/\tau) > 0$ by the group of translations generated by $T_\tau : w \mapsto w + \tau$. Being close to $p$ means $\text{Im}(w/\tau)$ is high. By shifting the boundary of $H$ to $\text{Im}(w/\tau) = h_0 > 0$ we may assume that $V_0 \subset V'$. For any $R > 0$, there exists $h > 0$ such that for all $w_0 \in \mathbb{C}$ with $\text{Im}(w_0/\tau) > h + h_0$, then $H$ contains the half-strip $S$ of equation $\text{Re}(w - w_0) < +\log R$ and $|\text{Im}(w - w_0)| < \pi$. The map $w \mapsto \exp(w - w_0)$ sends bijectively $S$ to the slit disk $B(0, R) - (-R, 0]$ and its inverse, followed by $\langle T_\tau \rangle$ and the isomorphism to a neighborhood of $p$, is the sought for affine immersion.

For an irregular singularity, the repelling petal in which the geodesic enters is isomorphic to a left half-plane in $\mathcal{E}$ and the proof is the same as above (without quotienting by a translation).

□

In an affine surface, a geodesic can be viewed as an $\mathbb{R}$-affine immersion of an interval $I \subset \mathbb{R}$. In the next lemmas, when these trajectories encounter singularities, we draw upon the local models to extend the immersions to a suitable open subset of $\mathbb{C}$ containing $I$.

**Lemma 4.10.** Given a meromorphic connection $\nabla$ on a Riemann surface $S$, consider a geodesic arc $\gamma$ defined on $(0, 1)$ and such that $\gamma(t)$ converges to a point $p \in S$ as $t \to 0$. Then, $\gamma$ extends to the affine immersion of a slit disk

$$C := \{z \in \mathbb{C}; 0 < |z| < \epsilon, -\pi < \arg(z) < \pi\}$$
for some $\epsilon > 0$. This immersion converges to $p$ at $z \to 0$.

**Proof.** By Lemma 4.1, $p$ is either regular, conical Fuchsian or irregular, in which case $\gamma$ tends to one of the foci of $p$ and let $p$ denote this focus instead. Apply Lemma 4.9 to this $p$ and $R = 2$: we get a neighborhood $V$ with some property. There exists $t_0 \in (0, 1)$ such that $\forall t \ge t_0$, $\gamma(t) \in V$. The property gives an affine immersion $f: \mathrm{Dom}(f) = B(0, 2) - (-2, 0] \to S$ sending 1 to $\gamma(t_0)$ and such that $f(z)$ tends to $p$ as $z$ tends to 0 within $\mathrm{Dom}(f)$. On the straight segment $[t_0, 0)$, the map $f_0: z \mapsto f(z/t_0)$ defines a geodesic from $\gamma(t_0)$ to $p$. By Lemma 2.13 and section 3.8, it must coincide with $\gamma$ on $[t_0, 0)$. The map $f_0$ provides the sought for immersion. $\square$

Note that the extension is not necessarily injective in the Fuchsian conical case.

**Lemma 4.11.** Given a meromorphic connection on a Riemann surface $S$, consider an affine immersion $\gamma: \mathbb{R}_{>0} \to S$ such that $\gamma(t)$ converges to some $p \in S$ as $t \to +\infty$. Then $p$ is a pole of $\nabla$ and $f$ extends to the affine immersion of an unbounded sector

$$C := \{z \in \mathbb{C}; |z| > R, -\epsilon < \arg(z) < \epsilon\}$$

for some $R > 0$ and $\epsilon > 0$. If $p$ is not a cylindrical Fuchsian singularity, we can take $\epsilon = \pi$. Moreover, all the geodesics $t \mapsto \gamma(te^{i\theta})$ for $-\epsilon < \theta < \epsilon$ converge to $p$ as $t \to +\infty$ (to the same swath if $p$ is a multiple pole).

**Proof.** By Lemma 2.12, the limit $p$ of $\gamma(t)$ can only be an irregular singularity, or a Fuchsian one, either cylindrical or anti-conical.

In the irregular case, Lemma 3.36 tells us the geodesic ends up, say for all $t > t_0$, in an attracting petal, which is isomorphic to the universal covering $\tilde{U}$ of $U$ where $U$ is complement in the canonical affine plane $\mathbb{C}$ of a closed disk of center 0. In $U$, the geodesic takes the form for $t > t_0$ of an affinely parametrized straight line $at + b$, and for $\epsilon$ small enough and $R$ big enough, the set $C$ of the statement, with $\epsilon = \pi$, is mapped injectively by $z \mapsto az + b$ to a single sheet of the universal covering $\tilde{U} \to U$.

In the cylindrical Fuchsian case, we end up in a model $\mathbb{H}/\mathbb{Z}$ with an affine line $t > t_0 \mapsto at + b$ whose imaginary part tends to $\infty$ and the argument is similar, but we need the image of $C$ by $z \mapsto az + b$ to remain contained in $\mathbb{H}$, so $\epsilon$ may need to be small.

In the anti-conical Fuchsian case, we distinguish two cases: in the pure case, we end up as in the proof of Lemma 4.10 in a model $H/\langle T_t \rangle$ in the exponential-affine plane $\mathcal{E}$ with this time $\mathrm{Im} t < 0$. In the coordinate $u = \exp w$, the geodesic take the form $t \mapsto at + b$. For $R$ big enough, the map $z \mapsto az + b$ lifts on the set $C$ with $\epsilon = \pi$ to a map $C \to H$ which projects under the quotient by $T_t$ to an affine extension of the geodesic.

In the shifted anti-conical case, given any geodesic $t \mapsto at + b$ in the model, we can embed the image of $C$ with $\epsilon = \pi$ by the extension $z \mapsto az + b$ provided $R$ big enough. $\square$

Actually in the non-cylindrical case, we can even take $\epsilon > \pi$ (a sector of opening angle bigger than a full turn, so that does not embed in $\mathbb{C}$).

**4.3. Cylinders and closed geodesics.** Given an affine surface $\mathcal{A}$, we define a closed geodesic as a geodesic that eventually comes back to its initial point with the same initial direction, but possibly a different speed.

Consider a closed geodesic $\gamma: I \to \mathcal{A}$, with $[a, b] \subset I$, $I \subset \mathbb{R}$ an open interval, such that, at $a$ and $b$, $\gamma$ has the same value and direction. Since the vectors $\gamma'(a)$ and $\gamma'(b)$ in the tangent space to $X^*$ at $\gamma(a) = \gamma(b)$ are related by $\gamma'(b) = s\gamma'(a)$ for some real $s > 0$.

**Note:** the holonomy of the parallel transport along $\gamma$ from $a$ to $b$ is the multiplication by $s$.

We saw in Section 2.3 that the supports of all closed geodesics of translation/dilation cylinders define a horizontal/radial foliation.
**Lemma 4.12.** There exists an immersed translation cylinder if $s = 1$, an immersed dilation cylinder if $s \neq 1$, such that $\gamma$ is a closed geodesic of this cylinder. More precisely $\gamma = \psi \circ \tilde{\gamma}$ where $\psi : C \to A$ is an affine immersion, $C$ is the cylinder, $\tilde{\gamma}$ is a closed geodesic of $C$.

*Proof.* By uniqueness of solution of regular O.D.E. (Cauchy-Lipschitz), $\gamma \circ \phi = \gamma$ holds on $I \cap \phi^{-1}(I)$, which contains $b$, where $\phi(t) = s(t-b)+a$, $\phi \in \mathrm{Aut}(\mathbb{C})$. If $s \neq 1$ the map $\phi$ has a unique fixed point $z_0$, it is real and cannot belong to $I$ (geodesics are locally injective). In particular, the sets $\phi^n([a, b])$ for $n \in \mathbb{Z}$ cover an open set containing $I$ and this is the case too if $s = 1$ as then they cover $\mathbb{R}$.

Consider an affine extension $\hat{\gamma} : U \to X^*$ where $U$ is an open subset of $\mathbb{C}$ containing $I$. If $s = 1$ let $S$ be an open rectangle

$$ \{z \in \mathbb{C}; \mathrm{Re}(z) \in (a - \epsilon, b + \epsilon), \mathrm{Im} z \in (-h, h)\} $$

contained in $U$. If $s \neq 1$ let $z_0$ be the fixed point of $\phi$ and $S_0$ a set of the form

$$ \{z \in \mathbb{C}; \mathrm{Re}(z) \in [a - \epsilon, b + \epsilon], \arg(z - z_0) \in (-\alpha, \alpha)\} $$

if $z_0 > b$ or

$$ \{z \in \mathbb{C}; \mathrm{Re}(z) \in [a - \epsilon, b + \epsilon], \arg(z - z_0) \in (\pi - \alpha, \pi + \alpha)\} $$

if $z_0 < a$. Let $C$ be the affine surface defined as the quotient of $S$ by the relation $z \sim \phi(z)$ and denote $\pi : S \to C$ the quotient map. The relation $\hat{\gamma} \circ \phi = \hat{\gamma}$ holds on $S \cap \phi^{-1}(S)$ by the affine identity theorem. We can thus factor $\hat{\gamma} = \psi \circ \pi$. We define $\tilde{\gamma}(t)$, first for $t \in [a - \epsilon, b + \epsilon]$ as $\pi(t)$, and extend it to $I$ is a geodesic of $C$ using $\phi$. $\square$

So $\gamma$ embeds into a one-parameter family of closed geodesics (up to reparametization) with the same holonomy factor $s$.

The closed geodesics of a cylinder can be parametrized by an interval in a way that is meaningful in affine geometry. In the case of a translation cylinder, this corresponds to the parametrization of a geodesic transverse to all the geodesics in the cylinder. In the case of a dilation cylinder, it corresponds to the angular parametrization of one of the concentric circular arcs of the annulus. In both cases, the interval can be finite, semi-infinite or infinite. If the one-parameter family of closed geodesics cannot be extended further, we say that the cylinder is maximal.

An infinite end of a cylinder corresponds to a Fuchsian singularity of residue $\mathrm{res}$ satisfying $\mathrm{Re}(\mathrm{res}) = 1$, see Section 2.4. We have $\mathrm{Im}(\mathrm{res}) = 0$ or not depending on whether the cylinder is a translation or dilation cylinder. Finite ends of cylinders will be described in Section 5.9.

Examples of self-intersecting attracting closed geodesics forming an immersed dilation cylinder are given in Section 3.3 and Figure 2 of [NST23].

**4.4. Anti-conical and swath domains.** Consider a finite type affine surface $(X, \nabla, S)$. We recall that saddle connections are finite length geodesics between points of $S$.

**Definition 4.13.** An *antic-conical domain* is a topological open disk $F$ in $X$, punctured at a unique point, which is an anti-conical Fuchsian singularity (i.e. of residue $\mathrm{res}$ satisfying $\mathrm{Re}(\mathrm{res}) > 1$) and such that $F$ is bounded by simple saddle connections that do not cross each other, is $m$-sided for some $m \ge 1$ and such that its $m$ corners have angles $\theta_1, \dots, \theta_m \ge \pi$.

A *swath domain* is a topological open disk $F$ containing no singularity, bounded by simple saddle connections that do not cross each others, such that $F$ is $m$-sided for some $m \ge 1$, with $m-1$ corners having angles $\theta_1, \dots, \theta_{m-1} \ge \pi$ while the remaining corner is an irregular singularity, of which a unique swath has a neighborhood included in $F$ (it makes sense to declare that this corner has infinite angle).
Remark 4.14. In the case of anti-conical domains, we have

$$ \sum_{k=1}^{m} (\theta_k - \pi) = 2\pi(\mathrm{Re}(\mathrm{res}) - 1). $$

as follows from Lemma 2.9.

**Lemma 4.15.** In an anti-conical or swath domain, any geodesic starting at a point of the boundary and pointing towards the interior converges in infinite time to the associated anti-conical Fuchsian singularity or swath.

We have in particular a form of concavity: no inner geodesic can link two points of the boundary.

*Proof.* Orient the sides of the face F counterclockwise (two sides on the same edge give the latter two opposite orientations).

<figure><img src="image_29.png" /></figure>

On this example, the turning angle $\alpha$ and the winding number $s$ are both negative.

If the signs of $\alpha$ and $s$ do not match, then following the geodesic back in time up to $\partial F$ we must cross $\partial U$ again.

<caption>FIGURE 29.</caption>

Consider a geodesic $\gamma: I \to X$ defined an interval $I$ such that $0 \in I \subset [0, +\infty)$, $\gamma(\mathrm{Int}) \subset F$ and $\gamma(0) \in \partial F$. We first prove that $\gamma$ cannot self-intersect. Otherwise, consider the minimal $a$ such that there is another $b > a$ for which $\gamma(a) = \gamma(b)$. We also take $b$ minimal. The simple closed loop $\gamma|_{[a,b]}$ cuts a topological disk $U$ out of the face. Let $s = 1$ if the restricted geodesic follows $\partial U$ it the positive (counterclockwise) orientation and $s = -1$ if it is the negative orientation: $s$ is also the winding number of the restricted geodesic in a topological chart sending $F$ to a disk in the plane). Let $\alpha$ be the turning angle of the geodesic at its origin. Note that $|\alpha| \in (0, \pi)$. The fact that the geodesic starts on $\partial F$ implies that $\alpha$ has the same sign as $s$. Indeed, if the signs did not match, following the geodesic in the past we would enter $U$ and would not be able to exit: see Figure 29). If $U$ does not contain a singularity, Lemma 2.9 implies $\alpha = s2\pi$, which contradicts $|\alpha| < \pi$. If $U$ contains the anti-conical point, Lemma 2.9 implies $\alpha = s2\pi(1 - \mathrm{Re}(\mathrm{res}))$ but the signs do not match since $1 - \mathrm{Re}(\mathrm{res}) < 0$.

We then prove that the geodesic cannot exit. This geodesic would cut the face into two topological disks, one of them, call it $F'$, does not contain the singularity or some swath neighborhood. Denote $\alpha, \beta \in (0, \pi)$ the angle of $F'$ at the ingress and egress points of the geodesic. Lemma 2.9 implies

$$ 2\pi = \pi - \alpha + \pi - \beta + \sum_{i \in I} \pi - \theta_i $$
where $I \subset \{1, \dots, m\}$ is the set of corners of $F$ still in $F'$. However, $\theta_i \ge \pi$ so the right hand side is $< 2\pi$, leading to a contradiction.

Similarly: (*) the geodesic cannot hit in finite time $\partial F$ on an apex (conical point or focus of an irregular singularity). Indeed the same contradiction would hold with the only difference that $\beta$ may or may not be $\ge \pi$.

An anti-conical singularity and a swath have a trapping neighborhood (see Lemma 2.14 and section 3.8), such that every geodesic entering it necessarily stays, and tends to the singularity or swath. By way of contradiction we assume that geodesic $\gamma$ never enters it. Consider the maximal interval of the form $(0, t_{\max})$ on which it is contained in $F$, with $t_{\max} \in (0, +\infty]$. Consider the accumulation set $K$ of $\gamma$ at $t_{\max}$. It is a compact and connected subset of $\bar{F}$. Hence either

(1) $K$ contains a non-singular point of $X$;

(2) $K$ is reduced to a singular point.

In the first case, using injectivity of $\gamma$ on $[0, t_{\max})$, we get again a contradiction from Lemma 2.9, as follows. Consider a local affine chart of the accumulation point whose image is a disk centered on the point. The traces of the geodesic in this disk are a sequence of straight chords that accumulate 0. The direction, taken modulo $\pi$, of such chords must converge. For any $\epsilon > 0$ we can find two consecutive chords on which the respective geodesic speed vectors make an angle $< \epsilon$ modulo $\pi$. By taking a short transversal segment between them we obtain a closed loop made of two pieces of geodesic and total turning number close to 0 or to $s/2$, where $s$ is as a few paragraphs above. It cannot be $-s/2$ for the same reason as previously. This contradicts Lemma 2.9 both in the case it contains an anti-conical singularity (the turning number must be $(1-\text{Re}(\text{res})s)$, and in the case it does not (the turning number must be $s$).

In the second case the geodesic tends to the singular point $p$, recall that: either $F$ contains an anti-conical singularity, no other singularity, and $\partial F$ contains only regular points and conical singularities; or $F$ contains a neighborhood of a unique swath, no singularity, and $\partial F$ contains only regular points, conical singularities and the irregular singularity of the swath. If $p$ is anti-conical we are done. If $p$ is a conical singularity, the local model of Section 2.4 implies that any geodesic tending to $p$ must hit it in finite time, contradicting (*). If $p$ is an irregular singularity, then by Lemma 3.36 either the geodesic tends to the swath and we are done, or it tends to a focus in finite time, again contradicting (*). $\square$

## 5. GEOMETRY OF AFFINE IMMERSIONS

The main goal of this section is to describe affine immersions of planes, open half-planes and open disks into a finite type affine surface, proving all the geometric results underlying the Delaunay decomposition constructed in Section 6. We more generally study immersions of convex subsets of $\mathbb{C}$.

We recall that the word affine is to be understood in this article as complex-affine. An affine map between affine surfaces is defined as a map that is locally affine in charts. It is necessarily an immersion and injective ones are necessarily embeddings. We choose here to use the terminology affine immersion to insist on the absence of injectivity hypothesis, and the terminology affine embedding by coherence.

We recall that marked points of a finite type affine surface $(X, \nabla, S)$ are erasable singularities, and stress that we decide to stop geodesics at marked points even though we may naturally extend them beyond.

### 5.1. Affine identity principle and relation between affine immersions.
We recall that the affine identity principle is the statement that if two affine maps are defined on the same connected subset of an affine surface $S$ and take values in some affine surface $S'$,
and if they coincide in the neighborhood of some point, then they are equal. An immediate consequence, bearing the same name, concerns two affine maps defined on two open subsets $U_1, U_2$ of $S$ with non-empty intersection and taking values in $S'$: if they coincide in the neighborhood of some point $s$, then they are equal on the connected component of $U_1 \cap U_2$ containing $s$. We will use several times the following consequence this principle.

<figure><img src="image_30.png" /><figcaption>FIGURE 30. Illustration of Lemma 5.1 in a case where the immersions are embeddings.</figcaption></figure>

**Lemma 5.1.** Let $S$ be an affine surface (not necessarily of finite type). Let $f_1: U_1 \to S$ and $f_2: U_2 \to S$ be two affine immersions with $U_1, U_2$ open subsets of the canonical affine plane $\mathbb{C}$. Assume that there exists $z_1 \in U_1$ and $z_2 \in U_2$ such that $f_1(z_1) = f_2(z_2)$. Consider the unique element $L \in \mathrm{Aut}(\mathbb{C})$ sending $z_1$ to $z_2$ and such that $f_1 = f_2 \circ L$ holds near $z_1$. Denote by $W$ be the connected component of $U_1 \cap L^{-1}(U_2)$ that contains $z_1$ and $W' = L(W)$ the connected component of $U_2 \cap L(U_1)$ that contains $z_2$. Then

$$f_1 = f_2 \circ L \text{ holds on } W.$$

In particular, $f_2(U_2) \cap f_1(U_1)$ contains $f_1(W) = f_2(W')$. See Figure 30.

For instance, in the particular case where $L^{-1}(U_2) \subset U_1$, we get $f_2(U_2) = f_1(L^{-1}(U_2)) \subset f_1(U_1)$.

We will apply this lemma several times. A typical situation is the following (we call strip any image by an element of $\mathrm{Aut}(\mathbb{C})$ of the set $\{z \in \mathbb{C}; \mathrm{Re}(z) \in (0, 1)\}$):

**Lemma 5.2.** If a geodesic $t \ge 0 \mapsto \gamma(t)$ tends at $+\infty$ to an anti-conical Fuchsian singularity $p$ of a meromorphic connection, for which we have a punctured neighborhood $V$ isomorphic to a standard neighborhood in a model of $p$ as in Section 2.4, and $\gamma$ extends to an affine immersion $f$ of a convex open set $\Delta \subset \mathbb{C}$ containing $[0, +\infty)$ and not equal to a strip between two parallel lines nor to the whole plane, then there is a neighborhood $W'$ of $\infty$ in $\Delta$ that is mapped to $V$.

The skew cylinder model of $V$ allows to control the immersion $f$ and the geodesics, in this part $W'$ of $\Delta$.

*Proof.* There is a punctured neighborhood $V$ of $p$ isomorphic: in the pure anti-conical case to a neighborhood of $\infty$ in a skew cylinder model $\tilde{C}^* \to \tilde{C}^* / \langle \lambda \rangle$; in the shifted anti-conical case, to a neighborhood of $\infty$ in the $n$-fold covering of the punctured canonical affine plane
<figure><img src="image_31.png" /><figcaption>FIGURE 31. Illustration of the proof of Lemma 5.2. The part of the skew cylinder that maps in V omits the semi-transparent area labeled A.</figcaption></figure>

C*, modified by a translation shift along a radial line. Let $t_0$ be such that $\gamma([t_0, +\infty)) \subset V$. In the model, this part of $\gamma$ lifts to a straight path $\tilde{\delta} : [t_0, +\infty) \to \tilde{C}^*$. This path $\tilde{\delta}$ projects to $C^*$ to a geodesic $\delta$ of the canonical affine plane $\mathbb{C}$. Let $L \in \mathrm{Aut}(\mathbb{C})$ be such that $\delta(t) = L(t)$. From the model, there is an affine immersion $f_2 : U \to V$ for a set of the form $U = \mathbb{C} \setminus (B(0, R) \cup D')$, where $D'$ is a half-line from 0 and $R$ big enough so that $L(\Delta) \cap U$ has one and only one connected component $W'$ (here we use that $\Delta$ is not equal to a strip between two parallel lines). It is necessarily unbounded and contains $\delta([t_1, +\infty))$ for some $t_1 \ge t_0$. See Figure 31. It follows that $W := L^{-1}(W')$ is a connected neighborhood of $\infty$ in $\Delta$. We then conclude with Lemma 5.1 applied to $f_1 = f$, $z_1 = t_1$, $U_1 = \Delta$, $f_2$, $z_2 = \delta(t_1)$ and $U_2 = U$ that $f = f_2 \circ L$ holds on $W = L^{-1}(W')$, so $f(W) \subset V$. $\square$

As in the case of analytic maps, there are stronger version of the affine identity theorem: for instance, two affine maps from a connected affine surface $S$ to an affine surface $S'$ to be equal, it is enough that they coincide on a set that has an accumulation point in $S$. We exploit this in the following statement.

**Lemma 5.3.** Let $f : \Delta \to S$ be an affine immersion of a convex subset $\Delta$ of $\mathbb{C}$. Let $z_0 \in \partial\Delta$ and $z_1 \in \Delta$. Assume that $f(z)$ converges in $X$ as $z$ tends to $z_0$ along the segment $(z_0, z_1]$, in other words, that the geodesic $t \mapsto f((1-t)z_0 + tz_1)$ converges at $t=0$. Denote $y$ this limit. Then either $y$ is a regular point (i.e. $y \in X^*$) and $f$ has an affine extension to a disk $B(z_0, r)$, or $y$ is a conical Fuchsian singularity or an irregular singularity and $f$ has an affine extension to a slit disk $B(z_0, r) \setminus I$ where the slit $I$ is a radius of the disk in the direction opposed to $z_1$. Moreover, $f$ extends continuously to $\{z_0\}$.

*Proof.* If $y$ is regular, by post-composing a chart near $y$ with an element of $\mathrm{Aut}(\mathbb{C})$ we can find a chart $\psi : V \to B(z_0, r)$ for some $r > 0$ and such that $\psi \circ f$ is the identity for every $z \in B(z_0, r) \cap (z_0, z_1]$. The domains of the maps $f$ and $\psi^{-1}$ both contained the convex, hence connected, set $B(z_0, r) \cap \Delta$, and coincide on $B(z_0, r) \cap (z_0, z_1]$, so are equal on $B(z_0, r) \cap \Delta$ by the version of the identity principle stated just before. We can thus extend $f$ with $\psi^{-1}$.

If $y$ is not regular then Lemma 4.1 implies it is a conical Fuchsian singularity, or an irregular singularity. Then Lemma 4.10 implies there is some $g : U := B(z_0, r) - I \to$ for some $r > 0$ and with $I$ a radius in the direction opposed to $z_1$. By the same argument as in the regular case, $f$ and $g$ must coincide on $\Delta \cap U = \Delta \cap B(z_0, r)$, so $g$ allows to extend $f$ as wished. Lemma 4.1 also states that $g$ converges to $y$ at $z_0$. $\square$

Independently, we state and prove below a useful result.
The affine surfaces classified in Proposition 5.5 have no apexes (conical singularity or focus of an irregular singularity), so no saddle connections. They form a subclass of the exceptional affine surfaces that we will classify in Section 6.6.

The following result uses techniques analogue to Proposition 5.5 and will be needed in the proof of Lemma 6.22 and theorem 1.6.

**Proposition 5.6.** Consider a finite type affine surface $(X, \nabla, \mathcal{S})$ such that there exists an affine immersion $f : \mathcal{E} \to X^*$. Then, either $X^*$ admits a whole plane immersion, or $(X^*, \nabla)$ has the exponential-affine plane $\mathcal{E}$ as a universal covering (hence is isomorphic to $\mathcal{E}$, a full Reeb cylinder, a skew cone or an affine torus).

*Proof.* As a Riemann surface $\mathcal{E}$ can be identified with $\mathbb{C}$. If $f$ is injective, then we can use Lemma 2.15 (an injective holomorphic map from $\mathbb{D}^*$ to a compact Riemann surface $X$ necessarily extends holomorphically at 0) to the map $z \in \mathbb{D}^* \mapsto f(1/z)$: we get that $f$ extends to a holomorphic map $\hat{f} : \mathbb{C}P^1 \to X$. The point $\hat{f}(\infty)$ is an irregular singularity (a pole of order 2 of the connection). It follows that $\hat{f}$ is injective. Holomorphic maps between compact Riemann surfaces are surjective. Hence $X^*$ is isomorphic to $\mathcal{E}$.

In the rest of the proof we assume that $f$ is non-injective. The affine surface $\mathcal{E}$ can be identified with the universal covering $\tilde{\mathbb{C}}^*$ of the punctured canonical affine plane $\mathbb{C}^*$. Let $(r_1, \theta_1)$ and $(r_2, \theta_2)$ be two points, in polar coordinates, mapped by $f$ to a same point $x_0 \in X^*$. Let $z_1 = r_1 e^{i\theta_1}$, $z_2 = r_2 e^{i\theta_2}$ and $\iota_1, \iota_2$ two local sections of $\pi : \tilde{\mathbb{C}}^* \to \mathbb{C}$ sending $z_i$ to $(r_i, \theta_i)$. There exists $\phi \in \mathrm{Aut}(\mathbb{C})$, uniquely determined by $(r_1, \theta_1)$ and $(r_2, \theta_2)$, such that $\phi(z_1) = z_2$ and $f \circ \iota_1 = f \circ \iota_2 \circ \phi$ holds near $z_1$.

Case 1: $\phi$ does not fix 0. Then either $0 \in \phi(B(z_1, |z_1|))$ or $0 \in \phi^{-1}(B(z_2, |z_2|))$. Up to permuting the roles of the $z_i$, we assume this is the first option. Let $\iota_1$ be the section on $B(z_1, |z_1|)$ of $\pi$ sending $z_1$ to $(r_1, \theta_1)$. By affine continuation, we get that $f = f \circ \iota_1 \circ \phi$ holds on $\pi^{-1}(\phi(B(z_1, |z_1|)))$. In particular, there is a point of $\tilde{\mathbb{C}}^*$ near which $f \circ \zeta = f$ where $\zeta : (r, \theta) \mapsto (r, \theta+2\pi)$ is a generator of the deck transformation group of $\pi$. By affine extension, the relation $f \circ \zeta = f$ holds on all $\tilde{\mathbb{C}}^*$. So $f$ factors through $\pi$ as $f = g \circ \pi$ for some affine immersion $g : \mathbb{C}^* \to X^*$ and with $\phi$ we can extend $g$ into some affine immersion $\hat{g} : \mathbb{C} \to X^*$.

Case 2: for any pair $(r_1, \theta_1)$ and $(r_2, \theta_2)$, the map $\phi$ fixes 0. Such a $\phi$ lifts to a global automorphism $\tilde{\phi} : \tilde{\mathbb{C}}^* \to \tilde{\mathbb{C}}^*$ sending $(r_1, \theta_1)$ to $(r_2, \theta_2)$ and by analytic continuation, $f \circ \tilde{\phi} = f$ holds on all $\tilde{\mathbb{C}}^*$. The set of all such $\tilde{\phi}$, together with the identity, then forms a subgroup $\Lambda$ of the automorphism group of $\tilde{\mathbb{C}}^*$ and this group must be discrete for otherwise it would contradict local injectivity of $f$. Hence the map $f$ factors to an embedding of the quotient manifold $\tilde{\mathbb{C}}^*/\Lambda$ in $X^*$ and $\Lambda$ is a lattice or a cyclic subgroup of the translation group of $\mathcal{E}$, so the quotient is a) an affine torus, b) a skew cylinder or c) a full Reeb cylinder. For a) the image $f(\mathcal{E})$ is compact, hence equal to $X$. In cases b) and c) we can proceed as in the injective case, where we used Lemma 2.15: for b) $X - f(\mathcal{E})$ consists in a conical and an anti-conical singularity and $X$ is a sphere, for c) $X - f(\mathcal{E})$ consists in two Reeb type singularities and $X$ is a sphere too. $\square$

5.3. **Boundary behavior.** We prove here that affine embeddings of open convex subsets of $\mathbb{C}$ in $X^*$ always extend to their boundary. Default of continuous extension of affine immersions are hence always related to their non-injectivity.

We start with the following lemma that concerns immersions (not only embeddings) and that will be useful later too.

**Lemma 5.7.** Consider an affine immersion $f$ of an open convex set $\Delta \subset \mathbb{C}$ into $X^*$. Let $z_\infty \in \partial\Delta$ and $z_0 \in \Delta$. Consider the semi-open segment $I = [z_0, z_\infty)$. Its image by $f$ is the support of a semi-open arc of geodesic. Let $y$ be an accumulation point of this geodesic,
i.e. $y = \lim f(z_n)$ with $z_n \in I$ and $z_n \to z_\infty$. Then $y$ cannot be a non conical Fuchsian singularity. If moreover $y \notin f(\Delta)$ then either:

(i) $y$ is regular and $f$ has an affine extension to a neighborhood of $z_\infty$;

(ii) $y$ is singular and $f$ has a continuous extension to $\Delta \cup \{z_\infty\}$.

In both cases, the extension maps $z_\infty$ to $y$.

*Proof.* Without loss of generality we may assume that $z_\infty = 0$ and $z_0 = 1$, so $I = [1, 0)$ and $z_n$ is a positive real. Let $r_0 \in (0, 1)$ such that $B(1, r_0) \subset \Delta$. Then for all $n > 0$, $B(z_n, r_0z_n) \subset \Delta$.

Note that $f$ restricted to $I$ is a geodesic. Assume that $y$ is a non conical Fuchsian singularity. By Lemma 2.14, $y$ has arbitrarily small trap neighborhoods, such that any geodesic entering the neighborhood through its boundary never leaves, but we get a contradiction because either it is impossible to reach the singularity in the case of a Reeb type one, or it takes an infinite amount of time to accumulate it. Thus $y$ can only be a regular point of $X$, a conical Fuchsian singularity or an irregular singularity.

If $y$ is an irregular singularity, there are also arbitrarily small traps, described in Section 3.8: they are the union of sepals and attracting petals. After crossing the boundary of this trap, a geodesic takes infinite time to accumulate the singularity. The complement of the traps consists in interpetal spaces that converge to the apexes (foci) of $y$. Hence a geodesic accumulating $y$ must accumulate the set of apexes. Up to taking a subsequence we can assume that $f(z_n)$ tends to one of the apexes. In this case we denote $P$ a repelling petal associated to this apex.

For any $R > 0$ by Lemma 4.9 (applied to $R + 1$ composing $\phi$ by $1-z$ and taking a restriction) there exists a punctured neighborhood $V$ of $y$ in case (1), a repelling petal neighborhood of the apex in case (2), such that $\forall x \in V$ and $\forall x \in V \cap P$, there is an affine immersion (possibly injective) $\phi$ from the slit disk

$$B(0, R) \setminus [1, R) = \mathrm{Dom}(\phi)$$

to $X^*$ sending $0$ to $x$ and such that $\phi(z)$ tends to $y$ as $z$ tends to $1$ within $\mathrm{Dom}(\phi)$. We choose $R = 2/r_0$ (note that $R > 2$) and consider such a $V$. There exists $n$ such that $f(z_n) \in V$. Consider the map $\phi$ associated to $x = f(z_n)$ as above. Let us apply Lemma 5.1 to

$$f_1 = \phi, z_1 = 0, U_1 = \mathrm{Dom}(\phi),$$

$$f_2 = f, z_2 = z_n \text{ and } U_2 = \Delta \cap B(z_n, 2|z_n|).$$

It involves some $L \in \mathrm{Aut}(\mathbb{C})$ such that $L(0) = z_n$ and that the identity $f_1 = f_2 \circ L$ holds on the connected component $W$ of $U_1 \cap L^{-1}(U_2)$. See Figure 32.

If $1 \in L^{-1}(U_2)$ then note that since $U_2$ is convex, the whole set $[0, 1)$ is contained in $L^{-1}(U_2) \cap \mathrm{Dom}\phi$ hence in $W$. Then $f = \phi \circ L^{-1}$ on the ray $L([0, 1))$, so it tends to $y$ along this ray, which is a semi-open segment from $z_n$ ending at some inner point of $U_2$. By continuity, $f$ takes the value $y$.

If $1 \notin L^{-1}(U_2)$, then since $R \ge 2/r_0$, the convex set $L^{-1}(U_2) = L^{-1}(\Delta \cap B(z_n, 2|z_n|))$ is contained in $B(0, R)$. It contains $0$ and is disjoint from the slit $[1, R)$, for otherwise, by convexity, it would contain $1$, contradicting our assumption. So $L^{-1}(U_2) \subset U_1$ and the identity $f = \phi \circ L^{-1}$ holds on $U_2$, which is the intersection of $\Delta$ with a neighborhood of $0$ in $\mathbb{C}$. This provides a continuous extension to $f$ at $0$, and actually an affine extension to a neighborhood of $0$ in $\mathbb{C}$ if either $L^{-1}(0) \ne 1$ or $y$ is regular (for, then, we can take $\phi$ that extends to the whole disk $B(0, R)$). $\square$

In Case (ii), the proof actually provides an extension to a slit neighborhood of $z_\infty$, but note that Lemma 5.3 also gives it.

Lemma 5.7 above allows us to draw an interesting consequence for embeddings of convex sets.
<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /><figcaption>FIGURE 32. Illustration of the proof of Lemma 5.7.</figcaption></figure>

**Corollary 5.8.** Any affine embedding $f$ of an open convex set $\Delta \subset \mathbb{C}$ into $X^*$ extends continuously to the boundary of $\Delta$ in $\mathbb{C}$.

*Proof.* Consider as in Lemma 5.7 a point $z_\infty \in \partial\Delta$, any $z_0 \in \Delta$ and the semi-open segment $I = [z_0, z_\infty)$. By compactness of $X$ there is an accumulation point $y$ of $f$ at $z_\infty$ along $I$, i.e. $y = \lim(z_n)$, $z_n \in I$, $z_n \to z_\infty$.

By Lemma 5.7 it is enough to prove that $y$ cannot belong to the image of $f$. Indeed, if there were $z \in \Delta$ such that $y = f(z)$, then consider a neighborhood $V$ of $z$ such that $z_n \notin V$ provided $n$ is large enough. Since $f$ is a local diffeomorphism in a neighborhood of $z$, we can find an element of $V$ that has same image as $z_n$ under $f$. This contradicts the injectivity of $f$ and proves the claim. $\square$
The continuous extension actually holds in a stronger sense: one can replace the range X of the extension by the union of X* and all the apexes (conical Fuchsian singularities and foci of irregular singularities), with the appropriate topology near foci described in Section 3.7.

We end with a general result concerning continuous extension of immersions of convex sets.

**Lemma 5.9.** Consider an affine immersion $f$ of an open convex set $\Delta \subset \mathbb{C}$ into $X^*$. Let $C_0$ be the set of points in $\partial\Delta$ where $f$ has a continuous extension. Then $C_0$ is an open subset of $\partial\Delta$, the extension maps in the set of regular points, conical Fuchsian and irregular singularities, the points mapped to non-regular points form a closed and discrete subset of $C_0$.

*Proof.* Denote $\hat{f}$ the extension. Consider $z_0 \in C_0$ and choose any $z_1 \in \Delta$: the conditions of Lemma 5.3 are in particular met, and the lemma tells us that $y := \hat{f}(z_0)$ cannot be a non-conical Fuchsian singularity. Moreover Lemma 5.3 tells us that there is an affine extension of $f$ to a neighborhood of $z_0$ or to a slit neighborhood of $z_0$, with the slit in the direction opposite to $z_1$. The extension region contains a neighborhood of $z_0$ in $\partial\Delta$, so $C_0$ is open and points mapped to non-regular points are isolated. Closedness follows from continuity. $\square$

**5.4. Immersed sectors around geodesics.** In this section, $(X, \nabla, S)$ is a finite type affine surface. For the affine immersion of an infinite convex open sector $\Delta$ of vertex $z_0$, we refer to a geodesic contained in $\Delta$ and starting at $z_0$ as a radial geodesic.

**5.4.1. Behavior at infinity.** After studying limits on boundary points we turn in the next two statements to limits at infinity. An infinite open sector

$$ \Delta = \{z \in \mathbb{C} \setminus \{z_0\} ; \arg(z - z_0) \in (\alpha, \beta)\} $$

is convex iff $|\beta - \alpha| \le \pi$. We call $(\alpha, \beta) \subset \mathbb{R}/2\pi\mathbb{Z}$ the angular span of $\Delta$. The first statement is about limits along geodesics.

**Lemma 5.10.** Consider an affine immersion $f$ of a convex open infinite sector $\Delta$ in the canonical affine plane $\mathbb{C}$ into $X^*$. Consider a geodesic $\gamma : t \in (0, +\infty) \to a_\gamma t + b_\gamma \in \Delta$ not parallel to $\partial\Delta$ and $y \in X$ an accumulation point as $t \to +\infty$ of the geodesic $t \in (0, +\infty) \mapsto f \circ \gamma(t)$ of $X$. Then either:

(i) $y$ is regular and $y \in f(\Delta)$;

(ii) $y$ is Fuchsian non conical and $f \circ \gamma$ tends to $y$ at infinity;

(iii) $y$ is irregular and $f \circ \gamma$ tends to a swath of $y$ at infinity.

*Proof.* By performing an affine change of variable on $\Delta$, we may assume that $\gamma(t) = t$. The sector $\Delta$ is based on some $z_0 \in \mathbb{C}$. Since the two boundary lines of $\Delta$ are non horizontal, there exists $r_0 > 0$ such that $\forall t > 0$, $B(t, r_0t) \subset \Delta$. We recall that no geodesic can accumulate a Reeb type singularity, so $y$ is not such a point.

Assume that $y$ is a regular point. Assume by contradiction that $y \notin f(\Delta)$. Consider a local chart $\phi$ defined on an open neighborhood $V$ of $y$, with $\phi(V) = B(\phi(y), r)$. Consider $t \in (0, +\infty)$ such that $f(t) \in \text{Dom}(\phi)$ and denote $z_1 = \phi(f(t))$. There is a unique $L \in \text{Aut}(\mathbb{C})$ such that $L(z_1) = t$ and $\phi^{-1} = f \circ L$ holds near $z_1$. This map $L$ may depend on $t$. We will apply Lemma 5.1 to

$$ f_1 = \phi^{-1}, z_1, U_1 = B(\phi(y), r) $$

$$ f_2 = f, z_2 = t \text{ and } U_2 = \Delta. $$

Denoting $W$ the connected component of $B(\phi(y), r) \cap L^{-1}(\Delta)$, we get that $\phi^{-1} = f \circ L$ holds on $W$ and particular that $f(\Delta)$ contains $\phi^{-1}(W)$. The convex set $L^{-1}(\Delta)$ cannot contain $\phi(y)$ for otherwise it would contain the segment $[\phi(f(t)), \phi(y)]$, so $W$ would contain it
too, hence $f(\Delta)$ would contain $y$. From $\phi(y) \notin L^{-1}(\Delta)$ and $B(t, r_0t) \subset \Delta$, we get that the scaling factor of $L^{-1}$ is at most $|\phi(f(t)) - \phi(y)|/r_0$. Hence each time $|\phi(f(t)) - \phi(y)| \times (1 + 2/r_0) < r$, then $L^{-1}(\Delta \cap B(t, 2t)) \subset B(\phi(y), r)$. Since this happens for arbitrarily high values of $t$, this implies that the whole set $L^{-1}(\Delta)$ is contained in $B(\phi(y), r)$. But $L^{-1}(\Delta)$ is unbounded, leading to a contradiction. So $y$ cannot be regular.

If $y$ is a conical Fuchsian singularity, then $y \notin f(\Delta)$ since the latter is contained in $X^*$. We get contradiction by a similar approach to the Case (i), by proving that for any neighborhood $V'$ of $y$, $f(\Delta) \subset V'$.<fnref n="20" /> We imitate the proof for a regular point: Lemma 4.9 gives us some neighborhood $V$ of $y$, which, for $t > 0$ such that $f(t) \in V$, allows us to replace $\phi^{-1}$ in our argument by an affine immersion $f_1 : B(0, R) \setminus (-R, 0] \to V' \setminus \{y\}$ such that $f_1(1) = f(t)$. As in the case of regular points, the choice of an $R > 1 + 2/r_0$ ensures that $\Delta \subset V'$. We again get a contradiction.

IF $y$ is a anti-conical Fuchsian or cylindrical Fuchsian then (ii) follows from Lemma 2.14.

If $y$ is an irregular singularity, then consider a trapping neighborhood of its swathes consisting of the union of the attracting petals and of the sepals. If the geodesic enters the trap, then it tends to a swath; If the geodesic never enters a trap, then it accumulates a focus of $y$. We then get as in the conical case that the whole set $\Delta$ would be contained in a repelling petal $\pi(A_n)$, contradicting that such petal contains no geodesic defined for an infinite time. $\square$

*Remark 5.11.* In Case (i) of Lemma 5.10, $f$ is non-injective: we can for instance use the same argument as in the proof of Lemma 5.7. Alternatively, we can apply the lemma to the restriction of $f$ to $\Delta_t = T_{\gamma(t)}(\Delta)$ where $T_v(z) = z + v$. This implies that $y \in f(\Delta_t)$ for all $t > 0$. By applying this to cones of smaller opening we can even ensure that there is a sequence $z_n$ where $\phi(z_n) = y$ such that $\arg(z_n)$ tends to $\arg(a_\gamma)$ (the direction of $\gamma$).

In the next statement, we extend Lemma 5.10 to limits along more sequences.

**Lemma 5.12.** We assume that $X^*$ is not a full translation cylinder $(\mathbb{C}/\mathbb{Z})$. Consider an affine immersion $f$ of a convex infinite open sector $\Delta \subset \mathbb{C}$ into $X^*$. We denote acc for "accumulation set". Then exactly one of the following holds:

(i) for any geodesic $\gamma : (0, +\infty) \to \Delta$ not parallel to $\partial\Delta$, $\text{acc}_{t \to +\infty} f \circ \gamma(t) \subset f(\Delta)$;

(iia) $f(z)$ tends to a cylindrical singularity $y \in S$ as $d(z, \partial\Delta) \to +\infty$;

(iib) $f(z)$ tends to an anti-conical singularity $y \in S$ as $|z| \to +\infty$;

(iic) $f(z)$ tends to a swath of an irregular singularity $y \in S$ as $|z| \to +\infty$;

*Note that, since X is compact, the accumulation set of a geodesic is never empty. In Case (i), every accumulation point belongs to $X^*$, i.e. is a regular point. The cases are thus mutually exclusive.*

*Remark 5.13.* In the case where $X^*$ is a full translation cylinder, if a radial geodesic $\gamma$ of the sector $\Delta$ is mapped to a closed geodesic of the cylinder, then radial geodesics that belong to distinct connected component of $\Delta \setminus \gamma$ converge to distinct ends of the cylinder.

*Proof of Lemma 5.12.* We assume that we are not in Case (i), i.e. that there exists a geodesic $\gamma : (0, +\infty) \to \Delta$, not parallel to $\partial\Delta$, and $y \in X$ an accumulation point of $f \circ \gamma(t)$ as $t \to +\infty$, such that $y \notin f(\Delta)$. By Lemma 5.10 either $y$ is an irregular singularity and $f \circ \gamma$ tends to one of its swathes, or $y$ is Fuchsian non conical and $f \circ \gamma$ tends to $y$.

Assume that $y$ is anti-conical Fuchsian. By Lemma 5.2, there is a neighborhood of $\infty$ in $\Delta$ that immerses in a neighborhood of $y$ that is isomorphic to the infinite end of a skew cylinder, possibly shifted. Any geodesic $\gamma$ in $\Delta$ ends up staying in this neighborhood and

<footnote n="20">²⁰This is absurd on two levels: $f$ would have to map to the intersection of all $V'$, which is $\{y\}$, but $f$ maps to $X^*$; actually, mapping $\Delta$ affinely in a small enough neighborhood of $y$ is already impossible since $y$ has neighborhoods that contain no geodesic defined for infinitely long time.</footnote>
$f \circ \gamma$ is mapped to a straight line in the skew cylinder model $\mathbb{C}^*/\langle\lambda\rangle$ or in the shifted model.
In both cases, the line tends to the infinite point of the model, so $\gamma(t) \to y$.

In the case $y$ is a cylindrical Fuchsian singularity, consider a punctured neighborhood $V$ isomorphic via $\phi : \mathbb{H}/\mathbb{Z} \to V$ to a semi-infinite translation cylinder. We start to apply Lemma 5.1 as in the anti-conical case, but here there is less angular room in the universal covering $\mathbb{H}$ of the model as there was for an anti-conical singularity, and if the sector $\Delta' := L(\Delta)$ contains a direction of argument in $(-\pi, 0)$, then it will not be contained in $\mathbb{H}$, even if we remove a bounded subset from $\Delta$. But in this case, denote $T(z) = z + 1$. Since $\forall z \in \Delta' \cap T^{-1}(\Delta')$, the map $f \circ L$ takes the same value at $z$ and $T(z)$, we can extend $f \circ L$ to an affine immersion $g : \mathbb{C} \to X^*$ such that $g \circ T = g$. By Proposition 5.5 and our assumption that $y$ is cylindrical, $X^*$ is isomorphic to $\mathbb{C}/\mathbb{Z}$. In the favorable case where $L(\Delta)$ has angular span contained in $(-\pi, 0)$, then we can conclude as in the anti-conical case, except that for the convergence of $f(z)$ to $y$ we need the Euclidean distance from $z$ to $\partial\Delta$ to tend to $\infty$ to treat the case where one of the two boundary lines of $\Delta$ would be horizontal in the cylinder $\mathbb{H}/\mathbb{Z}$.

The last case is when $y$ is an irregular singularity and $f \circ \gamma$ tends to a swath. In particular, it eventually enters, using the notations of Section 3.8, an attracting petal $\pi(B_{n+1/2})$ and its real part in $B_{n+1/2}$ tends to $\infty$. The set $B_{n+1/2}$ is a right half plane in the exponential-affine plane $\mathcal{E}$, and exp is an affine map on this set, realizing it as the universal covering of the complement of a disk in the canonical affine plane $\mathbb{C}$. We then conclude, with an argument similar to the anti-conical case, that $f(z)$ tends to the corresponding swath as $z \to \infty$ in $\Delta$. $\square$

From the proof of Lemma 5.12, we get the following precisions:

**Complement 5.14.** In Case (iia) of Lemma 5.12 above, there exists $L \in \operatorname{Aut}(\mathbb{C})$ such that the angular span of $L(\Delta)$ is contained in $(0, \pi)$ and such that $f = \psi \circ \pi \circ L$ holds on $L^{-1}(\mathbb{H}) \cap \Delta$, where $\pi : \mathbb{H} \to \mathbb{H}/\mathbb{Z}$ is the canonical projection and $\psi$ is an affine isomorphism from $\mathbb{H}/\mathbb{Z}$ to a neighborhood of $y$.

**5.5. Holonomy set of an affine immersion.** In the following, $f : \Delta \to X^*$ is the affine immersion of an open convex set $\Delta \subset \mathbb{C}$ to a finite type affine surface $(X, \nabla, \mathcal{S})$. If $f$ is not injective, for every pair of distinct points $x, y \in \Delta$ such that $f(x) = f(y)$, there is a complex affine automorphism that conjugates a neighborhood of $x$ with a neighborhood of $y$ in $\Delta$ while being compatible with $f$: in other words, there is $\phi \in \operatorname{Aut}(\mathbb{C})$ such that $y = \phi(x)$ and $f \circ \phi(z) = f(z)$ provided $z$ is close enough to $x$. Since $\Delta$ is assumed convex, the set $\Delta \cap \phi^{-1}(\Delta)$ is connected, and by affine identity principle, the relation $f \circ \phi(z) = f(z)$ holds on all $\Delta \cap \phi^{-1}(\Delta)$.

**Definition 5.15 (Holonomy set).** An automorphism $\phi \in \operatorname{Aut}(\mathbb{C})$ belongs to the holonomy set $\operatorname{Hol}(f)$ of $f : \Delta \to X^*$ if it satisfies the following properties:

* $\phi^{-1}(\Delta) \cap \Delta \neq \emptyset$;

* for any $z \in \phi^{-1}(\Delta) \cap \Delta$, $f \circ \phi(z) = f(z)$.

**Remark 5.16.** The holonomy set encapsulates the defect of injectivity of $f$: the sets of pair $(x, y)$ such that $f(x) = y$ is retrieved as the set of $(x, \phi(x))$ for $\phi \in \operatorname{Hol}(f)$ and $x \in \Delta \cap \phi^{-1}(\Delta)$. In particular $f$ is injective if and only if its holonomy set is reduced to the identity.

**Remark 5.17.** Though $\operatorname{Id}_{\mathbb{C}} \in \operatorname{Hol}(f)$ and $\forall \phi \in \operatorname{Hol}(f)$, $\phi^{-1} \in \operatorname{Hol}(f)$, the subset $\operatorname{Hol}(f)$ of $\operatorname{Aut}(\mathbb{C})$ does not have to be a subgroup. Given $\phi, \psi \in \operatorname{Hol}(f)$, a sufficient condition for $\psi \circ \phi \in \operatorname{Hol}(f)$ is that there exists $z \in \Delta$ such that $\phi(z) \in \Delta$ and $\psi(\phi(z)) \in \Delta$. This automatically holds if $\Delta = \mathbb{C}$.

We denote $\mathcal{T} \cong (\mathbb{C}, +)$ the translation subgroup of $\operatorname{Aut}(\mathbb{C})$. We denote $\mathcal{D}$ the subgroup of translations and dilations of $\operatorname{Aut}(\mathbb{C})$, i.e. of $\phi : z \mapsto az + b$ with $a > 0$ and $b \in \mathbb{C}$.
**Lemma 5.18.** If $\Delta$ has infinite inner radius (i.e. contains disk of arbitrarily big radii) then $\mathcal{T} \cap \text{Hol}(f)$ is a discrete subgroup of $\mathcal{T}$.

If $\Delta$ contains a sector, then $\mathcal{D} \cap \text{Hol}(f)$ is a discrete subgroup of $\mathcal{T}$.

*Proof.* Given two translations $T_u$ and $T_v$ in $\text{Hol}(f)$, the sufficient condition in Remark 5.17 $T_v \circ T_u$ to be in $\text{Hol}(f)$ applies to $z$ at the center of a ball contained in $\Delta$ and of radius $> |u| + |v|$. We thus have a group. If it were not discrete, considering a sequence $T_{u_n}$ in it, with $u_n \to 0$ and $u_n \neq 0$, we would contradict that $f$ is locally injective.

In the case of $\mathcal{D}$ the proof is similar. $\square$

This applies for instance to sectors.

*Remark 5.19.* We can extend $f$ to an affine immersion $\hat{f} : \Delta \cup \phi^{-1}(\Delta) \to X^*$ by setting $\hat{f}(z) = f(z)$ when $z \in \Delta$ and $\hat{f}(z) = f(\phi(z))$ if $z \in \phi^{-1}(\Delta)$. The definition of $\text{Hol}(f)$ ensures that these two values coincide on $\Delta \cap \phi^{-1}(\Delta)$. Note however that trying to extend to $\Delta \cup \phi^{-1}(\Delta) \cup \phi^{-2}(\Delta)$ may fail, because $(\Delta \cup \phi^{-1}(\Delta)) \cap \phi^{-1}(\Delta \cup \phi^{-1}(\Delta))$ may fail to be connected. Consider for instance $\Delta$ to be an open half-plane not containing 0 and $f$ to be a branch of $z \mapsto z^{5/2}$.

Given $\phi \in \text{Aut}(\mathbb{C})$, we introduce the notations $a_\phi$ and $b_\phi$ where $\phi : z \mapsto a_\phi z + b_\phi$.

**Lemma 5.20.** For any $\phi \in \text{Hol}(f)$, the following two properties hold:

* if $\phi$ has a fixed point in $\Delta$ then $\phi = \text{Id}_\mathbb{C}$;

* the coefficient $a_\phi$ cannot be a negative real number.

*Proof.* No element of the holonomy set different from the identity can fix a point $z$ of $\Delta$ because then we would find pairs of elements mapped by $f$ to the same point of $X^*$ in arbitrarily small neighborhoods of $z$, contradicting the fact that $f$ is an immersion.

We deduce that no element $\phi$ of $\text{Hol}(f)$ can be such that $a_\phi \in \mathbb{R}_{<0}$. Indeed, such an affine map would fix a unique point $z_0 \in \mathbb{C}$. The intersection $\Delta \cap \phi^{-1}(\Delta)$ is a non-empty convex set any for any $z \in \Delta \cap \phi^{-1}(\Delta)$, the fixed point $z_0$ would lie in the interior of segment $[z, \phi(z)]$ and therefore would be a point of $\Delta$, contradicting the previous statement. $\square$

## 5.6. Half-plane immersions.
We will completely classify half-plane immersions.

We start by showing that the existence of a translation in the holonomy set of a half-plane immersion drastically constrains the geometry of the immersion.

**Lemma 5.21.** Consider an affine immersion $f : H \to X^*$ of an open half-plane $H$ into $X^*$. If $\text{Hol}(f)$ contains a non-zero translation, then at least one of the following statements holds:

(i) $f$ is the restriction of an affine immersion of the whole plane;

(ii) $\text{Hol}(f)$ is the group generated by a non-zero translation $\phi$ satisfying $\phi(H) = H$ and $f$ quotients-out to an affine embedding of the semi-infinite translation cylinder $H/\langle\phi\rangle$ as a punctured neighborhood of a cylindrical Fuchsian singularity of $X$.

*Proof.* Note that a translation $\psi \in \text{Aut}(\mathbb{C})$ has vector parallel to $\partial H$ if and only if $\psi(H) = H$. For any translation $\psi \in \text{Hol}(f)$, if $b_\psi$ is not parallel to the boundary of $H$, then $\psi$ can be used to extend $f$ to an immersion $\hat{f}$ of the whole plane $\mathbb{C}$ satisfying $\hat{f} = \hat{f} \circ \psi$. Claim (i) holds in this case. Recall we denote $\mathcal{T}$ the translation subgroup of $\text{Aut}(\mathbb{C})$ and that, according to Lemma 5.18, $\mathcal{T} \cap \text{Hol}(f)$ is a (discrete) subgroup of $\text{Aut}(\mathbb{C})$. Let $\mathcal{T}(H) \subset \mathcal{T}$ be the subgroup of translations preserving $H$ and $\Lambda := \text{Hol}(f) \cap \mathcal{T}(H)$, which is also a discrete subgroup of $\text{Aut}(\mathbb{C})$.

In the rest of this proof we assume that Claim (i) does not hold: in particular, every translation in $\text{Hol}(f)$ preserves $H$, i.e. $\Lambda \subset \mathcal{T}(H)$. By hypothesis, $\text{Hol}(f)$ contains a non-zero translation, so $\Lambda$, as a non-trivial discrete subgroup of $\mathcal{T}(H)$, is cyclic. Without loss of generality, we will assume that $\phi$ is a generator of $\Lambda$.
Still under the hypothesis that Claim (i) does not hold, we now prove that every element $\psi \in \text{Hol}(f)$ is a translation. We saw in Lemma 5.20 that $a_\psi$ cannot be a negative number. It follows that the intersection of $H$ and $\psi^{-1}(H)$ is a sector, of angle $\le \pi$. Deep enough in the sector, there exists $z \in H$ such that $\phi(z)$ is also in the sector, so both $\psi(z)$ and $\psi(\phi(z))$ belong to $H$. Since the map $\psi \circ \phi \circ \psi^{-1} \in \text{Aut}(\mathbb{C})$, sends $\psi(z) \in H$ to $\psi(\phi(z)) \in H$ for all $z$ in an open set, it belongs to $\text{Hol}(f)$. Note that $\psi \circ \phi \circ \psi^{-1}$ is the translation $z \mapsto z + a_\psi b_\phi$. Since $\phi: z \mapsto z + b_\phi$ is the generator of $\Lambda$, we get that $a_\psi \in \mathbb{Z}^* = \mathbb{Z} - \{0\}$. As the same reasoning can be carried out with $\psi^{-1}$, we deduce that both $a_\psi$ and $\frac{1}{a_\psi}$ belong to $\mathbb{Z}^*$. By Lemma 5.20, $a_\psi$ cannot be a negative number, so $a_\psi = 1$. Hence $\text{Hol}(f) = \Lambda = \langle \phi \rangle$.

By Remark 5.16, $f(z) = f(z')$ if and only if $z' \in \Lambda z$, so $f$ quotients-out to an injective map through $H \mapsto H/\Lambda$. The affine surface $H/\Lambda$ is isomorphic to $\mathbb{H}/\mathbb{Z}$. As a Riemann surface, $\mathbb{H}/\mathbb{Z}$ is isomorphic to the punctured unit disk $\mathbb{D}^*$ via the exponential map. By Lemma 2.16 the factor map is an affine isomorphism from $\mathbb{H}/\mathbb{Z}$ to a punctured neighborhood of a cylindrical singularity of $X$. $\square$

Since the commutators of affine functions are translations, we now deduce from Lemma 5.21 that the elements of the holonomy set of a half-plane immersion commute.

**Corollary 5.22.** Consider an affine immersion $f: H \to X^*$ of an open half-plane $H$ into $X^*$. Then the elements of $\text{Hol}(f)$ pairwise commute.

*Proof.* If $\text{Hol}(f)$ contains a non-zero translation, each of the two alternatives stated in Lemma 5.21 implies that the elements of $\text{Hol}(f)$ pairwise commute (see Section 5.2 for details on affine surfaces admitting the immersion of a plane). In the rest of the proof, we will assume that $\text{Hol}(f)$ does not contain any translation.

We consider a pair $\phi, \psi$ of elements in $\text{Hol}(f)$. We already know from Lemma 5.20 that $a_\phi$ and $a_\psi$ cannot be negative real numbers. Up to replacing $\phi$ or $\psi$ by its inverse, we can assume that $\arg(a_\phi) \in [0, \pi)$ while $\arg(a_\psi) \in (-\pi, 0]$.

We recall that, given $\alpha, \beta \in \text{Hol}(f)$, a sufficient condition for $\beta \circ \alpha \in \text{Hol}(f)$ is that there exists $z \in H$ such that $\alpha(z) \in H$ and $\beta(\alpha(z)) \in H$.

By the restriction we put on $\arg(a_\phi)$ and $\arg(a_\psi)$, the maps $\phi, \psi$ satisfy the condition above so $\psi \circ \phi \in \text{Hol}(f)$. Similarly, $\psi^{-1} \circ \phi^{-1} \in \text{Hol}(f)$. Moreover, the argument of the linear factors of $\psi \circ \phi$ belongs to $(-\pi, \pi)$ and the argument for $\psi^{-1} \circ \phi^{-1} \in \text{Hol}(f)$ is the opposite, so again we can apply the criterion and deduce that the commutator $[\phi, \psi]$ belongs to $\text{Hol}(f)$. The argument of the linear factor of $[\phi, \psi]$ is 1, i.e. it is a translation. By hypothesis, the translation $[\phi, \psi]$ has to be trivial so $\phi$ commutes with $\psi$. $\square$

Every set of pairwise commuting elements of $\text{Aut}(\mathbb{C})$ is contained in a maximal commutative subgroup of $\text{Aut}(\mathbb{C})$, which are the group of translations or the fixator of some point of $\mathbb{C}$.

We get a complete classification of half-plane immersions in finite-type affine surfaces.

**Proposition 5.23.** Consider an affine immersion $f: H \to X^*$ of an open half-plane $H$ into $X^*$. Then either:

(i) $f$ is an embedding;

(ii) $X^*$ is the full translation cylinder $\mathbb{C}/\mathbb{Z}$, a translation torus $\mathbb{C}/\Lambda$ or an affine torus $\tilde{\mathbb{C}}^*/\Lambda$;

(iii) $f$ factors to an embedding through a cylindrical Fuchsian singularity standard neighborhood;

(iv) $f$ factors to an embedding through an anti-conical Fuchsian singularity standard neighborhood, of opening angle $< \pi$;

(v) $f$ factors to an embedding through a Reeb cylinder of angle $\pi$.

More precisely, denote by $\mathcal{T}$ the translation subgroup of $\text{Aut}(\mathbb{C})$. In each corresponding case:
(i) $\bullet$ Hol($f$) is reduced to the identity;

* as $z$ tends to $\infty$, $f(z)$ tends to an anti-conical singularity of opening angle $\ge \pi$
or to a swath of an irregular singularity;

(ii) Hol($f$) is respectively equal to a non-trivial cyclic subgroup of $\mathcal{T}$, a lattice of $\mathcal{T}$,
the following a subset of the projection to Aut($\mathbb{C}$) of a lattice of the translation
group acting on $\mathcal{E}: \{u^P \circ v^q; p, q \in \mathbb{Z}, p\theta_u + q\theta_v \in (-\pi, \pi)\}$, $u, v$ is a pair of
translations of $\mathcal{E}$ of vectors of imaginary part $\theta_u$ and $\theta_v$ such that the affine
torus $\tilde{\mathbb{C}}^*/\Lambda$ is isomorphic via $\exp(\mathcal{E}/\langle u, v \rangle)$;

* $f = \pi \circ \iota$ where $\pi$ is the quotient map $\mathbb{C} \to \mathbb{C}/\mathbb{Z}, \mathbb{C} \to \mathbb{C}/\Lambda$ or $\tilde{\mathbb{C}}^* \to \tilde{\mathbb{C}}^*/\Lambda$ and $\iota$ an affine bijection from $H$ to a half-plane in $\mathbb{C}$ or $\tilde{\mathbb{C}}^*$;

(iii) Hol($f$) is a non-trivial cyclic subgroup of $\mathcal{T}$;

* $f = g \circ \pi$ where $\pi: H \to H/\langle T \rangle$ is the quotient map and $T$ is a translation preserving $H$ (so $H/\langle T \rangle \cong \mathbb{H}/\mathbb{Z}$ is a semi-infinite cylinder) and $g$ is an affine embedding from $H/\langle T \rangle$ onto a neighborhood of a cylindrical Fuchsian singularity $p \in \mathcal{S}$;

(iv) Hol($f$) is of the form $\{\phi^n; -k < n < k\}$ for some $\phi: z \mapsto az+b \in \text{Aut}(\mathbb{C})$ with $\arg(a) \in (0, \pi)$ whose fixed point is not in $H$ and where $k = [\pi/\arg(a)]$;

* $f = g \circ \pi \circ \iota$ where $\iota$ is an affine embedding of $H$ in $\tilde{\mathbb{C}}^*$, $\pi: \tilde{\mathbb{C}}^* \to \tilde{\mathbb{C}}^*/\langle \tilde{\phi} \rangle$ is the quotient map where $\tilde{\phi}$ sends $(r, \theta)$ to $(|a|r, \theta + \alpha)$ where $\alpha$ is the representative in $(0, \pi)$ of $\arg a$ (so $\tilde{\mathbb{C}}^*/\langle \tilde{\phi} \rangle$ is a skew cone) and $g$ is an affine embedding from $\pi \circ \iota(H)$ onto a neighborhood of an anti-conical Fuchsian singularity $p \in \mathcal{S}$;

(v) Hol($f$) = $\langle\phi\rangle$ for some dilation $\phi: z \mapsto az + b \in \text{Aut}(\mathbb{C})$ with $a \in (0,1)$ whose fixed point does not belong to $H$.

* $f = g \circ \pi|_H$ where $\pi: H' \to H'/\langle\phi\rangle$ is the quotient map, $H'$ is the half-plane containing $H$ and whose boundary passes through the fixed point of $\phi$ (so $H'/\langle\phi\rangle$ is a Reeb cylinder of angle $\pi$) and $g$ is an affine embedding from $H'/\langle\phi\rangle$ into $X^*$;

Case (ii) has non-empty intersection with Cases (iii) and (v).

Finally, $f$ extends continuously at every point of $\partial H$, except, possibly, one point in
Case (v) or in Case (ii) if $X^*$ is an affine torus.

*Proof.* By Remark 5.16, *f* is an embedding if and only if Hol(*f*) is reduced to the identity. In this case, a half-line in *H* is mapped to the support of a geodesic *γ* of *X*. Consider an accumulation point *y* ∈ *X* of this geodesic as in Lemma 5.12. If *y* were regular, Remark 5.11 would give us that *f* is non-injective. So we are in one of the cases (iia), (iib) or (iic) of Lemma 5.12. Case (iia) where *y* is a cylindrical singularity is incompatible with *f* being injective: indeed by Complement 5.14 applied to Δ = *H*, there exists *L* ∈ Aut(*C*) such that *L*(*H*) is an upper half-plane and such that *f* ◦ *L*⁻¹ = ψ ◦ *π* holds on *L*⁻¹(*H*) ∩ ℍ where *π*: *C* → *C*/ℝ is the canonical projection; but then *f* cannot be injective, leading to a contradiction. In Case (iib) if the anti-conical singularity had an opening angle *α* strictly less than *π*, we would obtain a similar contradiction to injectivity: indeed by the proof of Lemma 5.2, there is a neighborhood *W* of ∞ in *H* such that on *W*, *f* factors as *φ* ◦ *π*<sub>λ</sub> ◦ *ι* ◦ *L* where *L* ∈ Aut(*C*), *ι*: *L*(*W*) → *C*<sup>*</sup> is a section of *π*<sub>0</sub>: *C*<sup>*</sup> → *C*<sup>*</sup>, *π*: *C*<sup>*</sup> → *C*<sup>*</sup>/⟨*λ*⟩ and *φ* is an affine isomorphism from a neighborhood of the infinite end of *C*<sup>*</sup>/⟨*λ*⟩ to an punctured neighborhood of *y*. However, *L*(*W*) is a half-plane minus a bounded set, and contains a neighborhood of ∞ in a sector of opening angle bigger than *α*. The map *π*<sub>λ</sub> ◦ *ι* is not injective on such a set.

In the case where $X^*$ is the full translation cylinder, a translation torus or an affine torus,
the fact that $\pi$ is a universal covering and $H$ simply connected implies that $f$ factors as
$\pi \circ \tilde{f}$. The map $\tilde{f}$ being locally affine, defined on a half-plane, and taking values in $\mathbb{C}$ or
$\tilde{\mathbb{C}}^*$, it must be injective. The determination of Hol($f$) in each case is then easy: in the
first two cases it must coincide with the deck transformation group of $\pi$. In the case of
an affine torus, let $\tilde{H} = \tilde{f}(H)$; the transformation $\mu$ of $\tilde{\mathbb{C}}^*$ associated to $u^p \circ v^q$ satisfies $\mu(\tilde{H}) \cap \tilde{H} \neq \emptyset$ if and only if the rotation part of $u^p \circ v^q$ lies in $(-\pi, \pi)$.

In the rest of this proof, except in the last paragraph, we assume that we are not in Cases (i) or (ii). The set $\mathrm{Hol}(f)$ in particular contains an element different from the identity. We saw in Corollary 5.22 and the paragraph after it that $\mathrm{Hol}(f)$ is either contained in the translation subgroup of $\mathrm{Aut}(\mathbb{C})$ or in the fixator of a point $z_0$ in $\mathbb{C}$ that does not belong to $H$.

In the first case, Lemma 5.21 tells us that $f$ factors to an embedding through a cylindrical Fuchsian singularity neighborhood and that $\mathrm{Hol}(f)$ is a non-trivial cyclic subgroup of $\mathcal{T}$.

In the second case, we first assume that $\mathrm{Hol}(f)$ only contains dilations (fixing $z_0$). By Lemma 5.18, $\mathrm{Hol}(f)$ is a discrete subgroup of $\mathrm{Aut}(\mathbb{C})$. Let $\phi \in \mathrm{Aut}(\mathbb{C})$ be a generator and denote $\Lambda = \langle \phi \rangle = \mathrm{Hol}(f)$. Then by Remark 5.16, $f(z) = f(z') \iff z \in \Lambda z'$, So $f(H)$ is isomorphic to the image of $H$ in $\mathbb{C} \setminus \{z_0\}/\Lambda$, which is equal to $S/\Lambda$ where $S$ is the half-plane containing $H$ and whose boundary passes through $z_0$. We get a Reeb cylinder of opening $\pi$.

Finally we assume that $\mathrm{Hol}(f)$ contains a similarity $s_0$ fixing $z_0$ but which is not a dilation. We pass to the exponential-affine plane $\mathcal{E}$ using the change of variable $z = z_0 + \exp(w)$, pulling back $H$ to a subset $U$ of $\mathcal{E}$ that is either a horizontal strip of height $\pi$ if $z_0 \in \partial H$ or a shape of equations $y_1 - \pi/2 < y < y_1 + \pi/2$ and $x - x_1 > \ln(1/\cos(y - y_1))$, denoting $w = x + iy \in \mathcal{E}$. Let

$$g = f \circ (z_0 + \exp).$$

The set $\mathrm{Hol}(f)$ is in bijective correspondence with a set, which we denote $\mathrm{Hol}(g)$, of translations $T_u : w \mapsto w+u$ of $\mathcal{E}$ with the property that $T_u \in \mathrm{Hol}(g) \iff (T_u^{-1}(U) \cap U \neq \emptyset$ and $g \circ T_u = g$ holds on this intersection (which is connected)), and we know that $g(w) = g(w')$ iff $\exists T_u \in \mathrm{Hol}(g)$ such that $T_u(w) = w'$.

If $\mathrm{Hol}(g)$ contains two elements $T_u$ and $T_v$ with $\mathbb{R}u + \mathbb{R}v = \mathbb{C}$, then $g$ can be extended to an affine immersion $\hat{g} : \mathcal{E} \to X^*$. It passes to the quotient to an affine immersion of an affine torus $Y$ to $X^*$: $\tilde{g} : Y \to X^*$. The image being both open and compact, we get that $X^* = X = \tilde{g}(Y)$. Since $Y$ and $X$ are compact and $\tilde{g}$ is a local diffeomorphism, it is a covering. Hence $\hat{g}$, which is defined on the simply connected set $\mathcal{E}$, is a universal covering of $X$. Its deck transformation group is a subgroup of the automorphism group of the affine surface $\mathcal{E}$, i.e. the group of translations, and contains $T_u$ and $T_v$. It is thus a lattice and $X$ is an affine torus.

Otherwise, $\mathrm{Hol}(g) \subset T_{\mathbb{R}u_0}$ for some $u_0 \in \mathbb{C}^*$ with $\mathrm{Im}(u_0) \neq 0$ since $s_0 \in \mathrm{Hol}(f)$. We can use any non-trivial $T_u \in \mathrm{Hol}(g)$ to extend $g$ to an affine immersion $\hat{g}$ defined on $\hat{U} := \bigcup_{n \in \mathbb{Z}} T_u^n(U)$, which contains a half-plane whose boundary is parallel to $\mathbb{R}u_0$. Then by Remark 5.17, $\mathrm{Hol}(\hat{g})$ is a subgroup of $\{T_{su_0}; s \in \mathbb{R}\}$. If it were a dense subgroup we would contradict local injectivity of $g$, so we have a cyclic subgroup. Let $v \in \mathbb{C}$ with $\mathrm{Im}(v) > 0$ such that $T_v$ generates $\mathrm{Hol}(\hat{g})$. Denote $\pi : \mathcal{E} \to \mathcal{E}/\langle T_v \rangle$ the canonical projection. Note that the image is a skew cylinder and that $\pi(\hat{U}) = \pi(U)$ is a neighborhood of its infinite end. Then $\hat{g} = \psi \circ \pi$ for some $\psi$ which is injective by property of $\mathrm{Hol}(\hat{g})$ and is an embedding to a neighborhood of a pure anti-conical singularity by Lemma 2.16. Finally, $f = \psi \circ \pi \circ (z_0 + \exp)|_{U}^{-1}$. Concerning $\mathrm{Hol}(f)$, it is the set of $z \mapsto z_0 + (z - z_0)e^{nv}$ where $n \in \mathbb{Z}$ is such that $T_v^n \in \mathrm{Hol}(g)$, which is the case iff $-\pi < n \times \mathrm{Im}(v) < \pi$.

In Case (i), Corollary 5.8 ensures that $f$ extends continuously to $\partial H$. In the other cases, given a point $z_0 \in \partial H$ and $\epsilon > 0$, the set $\Delta' = H \cap B(z_0, \epsilon)$ is convex. If $f$ is injective on $\Delta'$, we can deduce by applying Corollary 5.8 to the $f|_{\Delta'}$ that $f$ extends continuously at $z_0$. This applies at every point of $\partial H$ for Cases (iii) and (iv). In Case (v), this also applies to every point of $\partial H \setminus \{z_0\}$. Last, in Case (ii), the immersion is explicit and always extends
for the full translation cylinder and the translation tori, and in the case of an affine torus $\tilde{\mathbb{C}}^*/\Lambda$, the extension is possible on $\partial H \setminus \{0\}$. $\square$

**5.7. Sector immersions.** We start by a study of the behavior at infinity.

**Lemma 5.24.** Let $\Delta \subset \mathbb{C}$ be an infinite convex open sector of opening angle $\theta \in (0, \pi)$ and $f: \Delta \to X^*$ an affine immersion. Assume that $\mathrm{Hol}(f)$ contains an element $\phi$ whose scaling factor $a_\phi$ satisfies $0 < |\arg(a_\phi)| < \theta$. Then exactly one of the following occurs:

(i) $X^* = X$ is an affine torus;

(ii) there is a neighborhood $V'$ of $\infty$ in $\Delta$ on which $f$ quotients-out to the embedding of a standard neighborhood of an anti-conical singularity of opening angle dividing $|\arg a_\phi|$; $f(z) \to p$ as $|z| \to +\infty$.

More precisely in Case (ii), $f|_{V'} = g \circ \pi \circ \iota|_{V'}$ where $\iota$ is an affine embedding of $\Delta$ in $\tilde{\mathbb{C}}^*$, $\pi: \tilde{\mathbb{C}}^* \to \tilde{\mathbb{C}}^*/\langle \tilde{\lambda} \rangle$ and $g$ is an affine embedding from $\pi \circ \phi(\Delta)$ onto a neighborhood of $p$.

*Proof.* Let $\alpha = |\arg(a_\phi)| \in (0, \pi)$. The proof of Case (iv) of Proposition 5.23 adapts to this case, with the following precaution: we first restrict $f$ to a set $V'$ of the form $\Delta' - B(0, R)$ with $\Delta'$ a sector based on the fixed point of $\phi$ and opening angle $\theta' \in (\alpha, 2\alpha)$ and such that $V' \subset \Delta$. We get a factorization for the restriction, and we extend the factorization to one $f$ that holds in a neighborhood of $\infty$ in $\Delta$, by using the affine identity principle. (Alternatively we can use $V'$ to extend the restriction to a half-plane containing a neighborhood of $\infty$ in $\Delta$ and then apply Proposition 5.23 so as to conclude with the identity theorem.) $\square$

The lemma below can be deduced from its version for planes (Lemma 5.21).

**Lemma 5.25.** Consider an affine immersion $f: \Delta \to X^*$ of an infinite open convex sector $\Delta$ into $X^*$. Denote $z_0$ the apex of the sector and write $\Delta = T_{z_0}(\tilde{\Delta})$.

1. The set $\mathcal{T} \cap \mathrm{Hol}(f)$, of elements of $\mathrm{Hol}(f)$ that are translations, is invariant by composition, thus forms a subgroup of $\mathrm{Aut}(\mathbb{C})$. It is discrete.

2. If $\mathrm{Hol}(f)$ contains a non-zero translation, then at least one of the following statements holds:

(i) $f$ is the restriction of an affine immersion of the whole plane;

(ii) $\mathcal{T} \cap \mathrm{Hol}(f) = \langle \phi \rangle$ with $\phi = T_v$, $v \notin \tilde{\Delta}$ and $f$ quotients-out by $\langle \phi \rangle$ to an affine immersion $\psi$, i.e. $f = \psi \circ \pi$ holds on $\Delta$ where $\pi: \mathbb{C} \to \mathbb{C}/\langle \phi \rangle$ and $\psi$ is defined on $\pi(\Delta)$. Moreover let $H$ be the maximal half-plane contained in $\bigcup_{n \in \mathbb{Z}} \phi^n(\Delta)$; the immersion $\psi$ is injective in the semi-infinite cylinder $H/\langle \phi \rangle$ and sends it to a standard neighborhood of a cylindrical Fuchsian singularity of $X$.

*Proof.* Recall that we denote $\mathcal{T}$ the translation subgroup of $\mathrm{Aut}(\mathbb{C})$ and that by Lemma 5.18, $\Lambda := \mathcal{T} \cap \mathrm{Hol}(f)$

is a discrete subgroup of $\mathcal{T}$.

If the vector $v$ of a translation in $\Lambda$ belongs to $\tilde{\Delta}$, then we can use $f \circ T_{nv}$ with successive values of $n \in \mathbb{N}$ to extend $f$ to a whole plane immersion.

On the opposite, if no non-trivial element of $\Lambda$ has its vector $v$ that belongs to $\tilde{\Delta}$, then $\Lambda$ is cyclic ($\tilde{\Delta}$ intersects any lattice). Let $\phi_0 = T_{v_0}$ be a generator:

$$ \Lambda = \langle \phi_0 \rangle. $$

We can still define an extension $\hat{f}$ of $f$ using $f \circ T_{nv_0}$ for all $n \in \mathbb{Z}$, but its domain now is not the whole plane. It contains however a maximal half-plane $H$ with $\partial H$ parallel to $v_0$. See Figure 33. Note that $\hat{f}|_H$ and $f$ coincide on the non-empty convex set $H \cap \Delta$. We can apply Lemma 5.21 to $\hat{f}|_H$:
<figure><img src="image_35.png" /><figcaption>FIGURE 33. The initial sector $\Delta$ on which $f$ is defined is in dark gray. We use the translation $T_{v_0} \in \mathrm{Hol}(f)$ to define an extension $\tilde{f}$ that is defined on the light and dark gray set. It contains a maximal half-plane $H$ bounded by the dashed line.</figcaption></figure>

Either $\tilde{f}|_H$ is the restriction of an affine immersion $\tilde{f}$ of the whole plane. Since $f$ and $\tilde{f}$ are both defined on $\Delta$ and coincide on the non-empty open subset $\Delta \cap H$, they coincide on $\Delta$, so $\tilde{f}$ is an extension of $f$.

Or $\hat{\Lambda} := \mathrm{Hol}(\tilde{f}|_H)$ is the group generated by a non-zero translation $\phi$ satisfying $\phi(H) = H$, i.e. of vector $u$ parallel to $v_0$, and $\tilde{f}|_H$ quotients-out to an affine embedding of the semi-infinite translation cylinder $H/\langle\phi\rangle$ as a punctured neighborhood of a cylindrical Fuchsian singularity of $X$. Note that $\phi$ is necessarily in $\mathrm{Hol}(f)$, i.e. in $\Lambda$, because the open set $(\Delta \cap H) \cap \phi^{-1}(\Delta \cap H)$ is non-empty. So $u \in v_0\mathbb{Z}$. Conversely, $\phi_0 \in \hat{\Lambda}$ because $(\Delta \cap H) \cap \phi_0^{-1}(\Delta \cap H)$ is non-empty. So $v_0 \in u\mathbb{Z}$. It follows that

$$ \hat{\Lambda} = \langle \phi_0 \rangle = \Lambda. $$

Note that $\Delta \setminus H$ has no two points related by $\langle\phi_0\rangle$; it follows that $f$ quotients-out by $\hat{\Lambda}$ on the whole set $\Delta$. $\square$

We have an analogue of Proposition 5.23, but weaker in that some of the statements only hold near $\infty$.

**Proposition 5.26.** Consider an affine immersion $f: \Delta \to X^*$ of an infinite open convex sector $\Delta$ of opening angle $\theta \in (0, \pi]$ into $X^*$. Then either:

(i) $f$ is an embedding in a neighborhood of $\infty$ and $f(z)$ tends, as $|z| \to +\infty$ to an anti-conical Fuchsian singularity of opening angle $\ge \theta$ or to a swath of an irregular singularity;

(ii) $X^*$ is the full translation cylinder $\mathbb{C}/\mathbb{Z}$, a translation torus $\mathbb{C}/\Lambda$ or an affine torus $\tilde{C}^*/\Lambda$; $f$ factors as $f = \pi \circ \phi$ where $\phi$ is an affine embedding of $\Delta$ in $\mathbb{C}$ or $\tilde{C}^*$;

(iii) $f$ quotients-out near $\infty$ by some $\phi_* \in \mathrm{Aut}(\mathbb{C})$ to an embedding through a cylindrical Fuchsian singularity standard neighborhood;

(iv) $f$ quotients-out near $\infty$ by some $\phi_* \in \mathrm{Aut}(\mathbb{C})$ to an embedding through a standard neighborhood of an anti-conical Fuchsian singularity $p$ of opening angle $\le \theta$; $f(z) \to p$ as $|z| \to +\infty$;

(v) $f$ quotients-out near $\infty$ by some $\phi_* \in \mathrm{Aut}(\mathbb{C})$ to an immersion through a dilation cylinder of opening angle $\ge \theta$.

In (v) the cylinder could very well be non-embedded (only immersed): see Section 4.3.

*Proof.* Case (i) is treated completely similarly as in the Proposition 5.23, as it involves Lemma 5.12, remark 5.11, and complement 5.14 which are all valid on infinite open convex sectors and Lemma 5.2 which is valid on unbounded open convex sets.
Case (ii) is treated as Case (ii) of Proposition 5.23. In the rest of this proof we assume
that we are not in Case (ii).

If there is some $\phi \in \text{Hol}(f)$ such that $\arg(a_{\phi}) \in (-\theta, 0) \cup (0, \theta)$, then by Lemma 5.24 we
are in particular in Case (iv). In the rest of this proof we assume we are not in this case
either, so in particular that for all $\phi \in \text{Hol}(f)$, either $\arg a_{\phi} = 0$ or $|\arg(a_{\phi})| \ge \theta$.

In Proposition 5.23, $\Delta = H$ was a plane and we could invoke Corollary 5.22, but here $\Delta$ is any sector and we proceed differently. One annoying hypothetical situation is illustrated on Figure 34. However we will see that there is not enough room in a finite type affine surface for this kind of immersions to exist. Consider any radial geodesic $\gamma$ of $\Delta$, say the one whose direction bisects the sector, and apply Lemma 5.10: either 1. the accumulation set of $f \circ \gamma(t)$ as $t \to +\infty$ is contained in $f(\Delta)$, or 2. $f \circ \gamma(t)$ tends to a singularity, which is 2.a Fuchsian non conical or 2.b irregular (in the latter case it tends to a swath).

<figure><img src="image_36.png" /><figcaption>FIGURE 34. A hypothetical sector in $\mathbb{C}$, whose immersion $f$ in an affine surface would have infinitely many non-identity elements $\phi \in \text{Hol}(f)$, all with disjoint ranges and domains, and no other ones. This situation in fact cannot happen. One could effectively define an affine surface $S$ by quotienting $\Delta$ by $\phi_1, \dots, \phi_n$ but it is not of finite type, and actually cannot embed in a finite-type affine surface as will follow from Proposition 5.26.</figcaption></figure>

If 2. above holds, then we proceed as in Lemma 5.12: there is a neighborhood of $\infty$ in $\Delta$ that immerses into a neighborhood of $y$ that is isomorphic to the infinite end of a skew cylinder in case 2.a by Lemma 5.2, or into a swath neighborhood isomorphic to a right half-plane in $\mathcal{E}$ in case 2.b. In case 2.b, $f$ is thus actually injective in a neighborhood of $\infty$ hence we are in Case (i). This is also the case in case 2.a if the opening angle $\theta$ of the skew cone is $< |\arg(a_\phi)|$. By hypothesis, we excluded the case where $\theta > |\arg(a_\phi)|$. If $\theta = |\arg(a_\phi)|$ we may be either in Case (iv) or in Case (i).

If 1. above holds, then consider an accumulation point $y$ of $f \circ \gamma$ and a small neighborhood $V$ around $y$ and an affine isomorphism $\psi: V \to D = B(0,1)$, with $\phi(y) = 0$. Branches of the geodesic in $V$ accumulate on $y$, so we can extract a sequence of branches whose direction converges to some limit direction. For each such branch there exists (apply Lemma 5.1) a subset of $\Delta$ mapped by $\psi \circ f$ to the intersection of $D$ with a sector of opening $\theta$, bisected by the branch, and based on one of the endpoints of the branch, see Figure 35. As soon as the branch passes close enough to $y$, we know that the intersection above will contain 0. Hence $y$ has arbitrarily large preimages $z_n$ by $f$. Moreover since the branches have a convergent direction, if we take $m$ and $n$ big enough, the map $\phi \in \text{Hol}(f)$ such that $\phi(z_m) = z_n$ satisfies that $\arg(a_\phi)$ is close to 0. From our standing assumption, we deduce
that $\arg(a_\phi) = 0$, so $\phi$ is a translation or a dilation fixing some point $z_1$ outside of $\Delta$ (see Lemma 5.20). We choose such $m$ and $n$ and denote $\phi_*$ the element $\phi$ constructed above.

<figure><img src="image_37.png" /><figcaption>FIGURE 35. The disk $D$ in the proof of Proposition 5.26; in red, branches of the geodesic; in gray, images of the sector $\Delta$ around two of these branches, by the affine maps $L \in \operatorname{Aut}(\mathbb{C})$ such that locally $\psi \circ f \circ \gamma = L$. When the branch is close to the center $0$, then $0 \in L(\Delta)$.</figcaption></figure>

The segment $[z_m, z_n]$ is entirely contained in $\Delta$ and is mapped by $f$ to a closed geodesic, because $\phi'_* \in \mathbb{R}_{>0}$, that belongs to some immersed translation or dilation cylinder (see Section 2.3). If $\phi_*$ is a translation, then we are in the situation of Lemma 5.25 and Claim (iii) holds (we cannot be in the situation (i) of Lemma 5.25 because then by Proposition 5.5 $X^*$ is a full translation cylinder or a translation torus, which we exclude by hypothesis, or a translation plane, in which case $f$ is injective, and we already covered this case).

<figure><img src="image_38.png" /><figcaption>FIGURE 36. Case $\phi_*$ is a dilation of factor 2 in the proof of Proposition 5.26; $\Delta$ is in dark gray; $S = T_{z_1}(\vec{\Delta})$ in blue; $\bigcup_{n \in \mathbb{Z}} \phi_*^n(\Delta)$ in light gray, dark gray and blue; a fundamental domain of a Reeb cylinder of angle $\pi$ in red.</figcaption></figure>

If $\phi_*$ is a dilation, recall that its fixed point $z_1$ does not belong to $\Delta$ and note that given $z \in \Delta$, the set of $k \in \mathbb{Z}$ such that $\phi_*^k(z) \in \Delta$ is an integer interval, i.e. of the form $\mathbb{Z}/\cap[a, b]$ (this follows from $\Delta$ being convex and the orbit being on a line, ordered like the exponent
k). From this, $f$ factors as $f = \psi \circ \pi$ where $\pi : \mathbb{C} \setminus \{z_1\} \to Y = (\mathbb{C} \setminus \{z_1\})/\langle\phi_*\rangle$ is the quotient to a dilation torus and $\psi : \pi(\Delta) \subset Y \to X^*$ is an affine immersion. The set $\Delta$ is actually contained in a sector $S_0$ of angle $\le \pi$ based on $z_1$, so $\pi(\Delta)$ is contained in a dilation cylinder of angle $\le \pi$. Moreover, denote $\Delta = T_{z^*}(\bar{\Delta})$ where $z^*$ is the tip of the sector $\Delta$ and let $S = T_{z_1}(\bar{\Delta})$. Then $\pi(\Delta)$ contains $\pi(S)$. Finally if we call $S'$ the biggest sector based on $z_1$ such that $\pi(S') \subset \pi(\Delta)$, then $\pi(S')$ is a dilation cylinder contained in the domain of $\psi$ and $\Delta \cap S'$ is a neighborhood of $\infty$ in $\Delta$. So Claim (v) holds. See Figure 36. □

**Complement 5.27.** Moreover, in Cases (iii) and (iv), if an element $\phi \in \text{Hol}(f)$ satisfies that $|\arg(a_\phi)| < \theta$, then $\phi$ is a power of $\phi_*$.

In Case (v) we can choose $\phi_*$ so that this is the case too. In Case (iv) we have moreover that $k|\arg(a_{\phi_*})| \le \theta$.

*Proof.* The convex set $\phi(\Delta) \cap \Delta$ contains a sector, thus is unbounded. So $\forall R > 0, \exists z \in \Delta$ such that $\phi^{-1}(z) \in \Delta$. By definition of $\text{Hol}(f)$, $f(\phi^{-1}(z)) = f(z)$ and this holds for nearby $z$ too.

In Case (iii) consider the decomposition $f = \psi \circ \pi$ of Lemma 5.25, where $\pi : \mathbb{C} \to \mathbb{C}/\langle T_v \rangle$ and $\psi$ is defined in $\pi(\Delta)$ and injective in a half-plane $H$. This injectivity means that $z$ and $\phi^{-1}(z)$ are related by a power of $T_v$, and this holds locally hence $\phi \in \langle T_v \rangle$.

In Case (iv) we use the decomposition $f|_{V'} = g \circ \pi \circ \iota|_{V'}$ of Lemma 5.24 where $g$ is injective near infinity. This implies similarly that $\iota(z)$ and $\iota(\phi^{-1}(z))$ are related by a power of $\lambda$, and since this holds locally $\iota \circ \phi \circ \iota^{-1}$ is the restriction of a power of $\lambda$. This power $k \in \mathbb{Z}$ is independent of $z$ and by letting $z$ tends to infinity we get that $k|\arg(a_{\phi_*})| \le \theta$. Note that the only dilation in $\text{Hol}(f)$ is the identity.

Last, in Case (v), the set of dilations that fix $z_1$ and belong to $\text{Hol}(f)$ is a discrete subgroup of $\text{Aut}(\mathbb{C})$ (it is a subgroup of $\mathcal{D} \cap \text{Hol}(f)$ using the notations of Lemma 5.18). We let $\phi_*$ be a generator. The hypothesis that $\phi(\Delta) \cap \Delta$ is unbounded means in particular that $|\arg(a_\phi)| \le \theta$. If $\arg(a_\phi) \ne 0$ then we know from the proof of Proposition 5.26 that we are in Case (v) of the proposition, for which we already determined that no dilation can be in $\text{Hol}(f) \setminus \{\text{Id}\}$. If $\arg(a_\phi) = 0$, recall that the dilations in $\text{Hol}(f)$ form a discrete subgroup of $\text{Aut}(\mathbb{C})$. If the fixed point of $\phi$ were different from that of $\phi_*$, by conjugating $\phi$ with powers of $\phi_*$ we would get a sequence of dilations in $\text{Hol}(f)$ not equal to $\text{Id}$ but tending to $\text{Id}$, contradicting discreteness. Hence the fixed points are the same, and $\phi$ is a power of the generator $\phi_*$. □

We deduce a description of the behavior at infinity of radial geodesics of open convex sector immersions. The following result extends Lemma 3.3 of [Tah23], originally established for dilation structures, to the broader setting of complex affine structures.

**Corollary 5.28.** Assume that $X^*$ is not a full translation cylinder, a translation torus, or an affine torus. Let $\Delta$ be an infinite open convex sector $\Delta$ and denote $\mathcal{R}$ the set of radial $\ell^{21}$ geodesics $\gamma : (0, +\infty) \to \Delta$. Given an affine immersion $f : \Delta \to X^*$, one of the following statements holds:

(i) there is an immersed dilation cylinder $C$ such that $\forall \gamma \in \mathcal{R}$, the geodesic $f \circ \gamma$ accumulates, as time goes to infinity, on a (possibly self-intersecting) attracting closed geodesic of $C$;

(ii) there exists a cylindrical Fuchsian singularity $y \in S$ such that $\forall \gamma \in \mathcal{R}$, the geodesic $f \circ \gamma$ converges to $y$ as times goes to infinity;

(iii) there exists an anti-conical singularity $y \in S$ such that $\forall \gamma \in \mathcal{R}$, the geodesic $f \circ \gamma$ converges to $y$ as times goes to infinity;

(iv) there exists an irregular singularity swath $s$ such that $\forall \gamma \in \mathcal{R}$, the geodesic $f \circ \gamma$ converges to $s$ as times goes to infinity.

<footnote n="21">²¹I.e. starting from the tip.</footnote>
**5.8. Immersions that do not extend continuously to a vertex.** The main difficulty in attempting to generalize the Delaunay construction to affine structures lies in the fact that they are not necessarily geodesically complete. In other words, a geodesic may blow up in finite time without hitting a singularity. This phenomenon is particularly apparent in a dilation cylinder, whose radial geodesic foliation has this property, see Section 2.3. In this section, we prove a partial converse.

**Proposition 5.29.** Consider an affine immersion $f$ of an open Euclidean triangle $\Delta$ to $X^*$ that does not extend continuously to one of its vertices $z_0$. Then $f$ extends to the immersion of the smallest infinite open convex sector $S$ of vertex $z_0$ containing $\Delta$. Moreover, one of the following statements holds:

(i) $X$ is an affine torus;

(ii) there exists a dilation $\phi_*$ fixing $z_0$ such that $f$ quotients to an immersion of the dilation cylinder $S/\langle\phi_*\rangle$, i.e. $f = \psi \circ \pi$ where $\pi: S \to S/\langle\phi_*\rangle$ is the quotient map; in particular, every radial geodesic of $S$ is mapped by $f$ to a (possibly self-intersecting) attracting closed geodesic of the cylinder.

*Proof of Proposition 5.29:*

*Construction of a special sequence* $\phi_n \in \mathrm{Hol}(f)$. We start by proving that there exists a sequence $(\phi_n)_{n \in \mathbb{N}}$ of elements of $\mathrm{Hol}(f)$ satisfying the following conditions:

$$ (1) \quad \lim_{n \to +\infty} a_{\phi_n} = 0; $$

$$ (2) \quad \lim_{n \to +\infty} \arg(a_{\phi_n}) = 0; $$

$$ (3) \quad \text{there exists sequences } x_n, x'_n \in \Delta \text{ such that } x_n \to 0, x'_n \to 0 \text{ and } \phi_n(x'_n) = x_n. $$

Consider a radial geodesic segment $\gamma: (0, 1] \to \mathbb{C}$ from $z_0$ to a point of the triangle $\Delta$, bisecting the angle of the triangle at $z_0$. Denote $\alpha = \arg \gamma'$ the direction of this bisector. Following Lemma 5.7, there exists $y \in f(\Delta) \subset X^*$ such that $f \circ \gamma(t)$ accumulates on $y$ as $t$ tends to 0.

The argument is then an elaboration on an argument in the proof of Lemma 5.7. Consider a neighborhood $V$ of $y$ affine isomorphic to the disk $D$ of center 0 and radius 1 via $\psi: V \to D$. Below we identify $V$ with $D$. The geodesic $f \circ \gamma$ cannot tend to $y$ nor to a point of $V$, for otherwise it would be possible to extend $f$ continuously at $z_0$. In particular the geodesic will enter and exit $V$ infinitely many times. Each time the geodesic $f \circ \gamma$ meets $V$, it does so along a chord of $D$, except possibly for one part which begins in $D$ corresponding to $t=1$. We get a sequence of chords $(A_k, B_k) = \psi \circ f(I_k)$, $I_k = \gamma((a_k, b_k))$, $(a_k, b_k) \subset (0, 1]$ disjoint intervals. The chords can be oriented by the parametrization by $t$. We know they accumulate on 0 (i.e. $y$). We can select a subsequence such that the oriented direction of the chord converges too. For each such chord $(A_k, B_k)$, there is $L_k \in \mathrm{Aut}(\mathbb{C})$ such that $\psi \circ f = L_k$ holds on $I_k$. The chord length tends to the diameter of $D$ while the length of the interval $(a_k, b_k)$ tends to 0. As a consequence, the constant $|L'_k|$ tends to $+\infty$ as $k \to +\infty$. The image $L_k(z_0)$ of the triangle vertex cannot belong to $D$ since $0 \notin (a_k, b_k)$. This implies that for all $k$ big enough, $L(\Delta)$ contains the center 0 of $D$. Let

$$ x_k = L_k^{-1}(0). $$

We just saw that $x_k \in \Delta$ for all $k$ big enough. Note that

$$ x_k \to z_0: $$

indeed $d(x_k, [\gamma(a_k), \gamma(b_k)]) = d(0, [A_k, B_k])/|L'_k| \le 1/|L'_k| \to 0$ and $\gamma(a_k)$ and $\gamma(b_k)$ tend to $z_0$. Since $x_k \ne z_0$ ($z_0$ is in $\Delta$ but not $x_k$) we can extract an injective subsequence, and assume $x_k \in \Delta$ for all $k$. Moreover, $f(x_k) = y$, so for any two terms $x_k, x_n$ of the sequence, there exists $\phi = \phi_{n,k} = L_n^{-1} \circ L_k \in \mathrm{Hol}(f)$ such that $\phi(x_k) = x_n$. By selecting $n$ and $k$ big, we can ensure that $\arg(a_\phi)$ is close to 0. By selecting $k$ first and then $n$ big enough, we
can ensure that $a_\phi$ is close to 0. We thus get a sequence $\phi_n$ satisfying the three required conditions by letting $x'_n = x_k$.

**Extension of $f$ to $S$.** Denote $S_0 = T_{-z_0}(S)$: it is an infinite sector whose tip is 0. For $n$ big enough, the triangle $\Delta$ contains the bounded sector $T_{x_n}(C')$ where $C' = S_0 \cap B(0, r)$, for some uniform $r > 0$. The triangle $\phi_n^{-1}(\Delta)$ thus contains the bounded sector $\{x'_n + a_{\phi_n}^{-1}z; z \in C'\}$ whose tip is close to $z_0$ and whose radius is $r/|a_{\phi_n}|$, hence large. It follows that for every compact subset $K$ of $S$, there exists $N > 0$ such that $\forall n \ge N$, $K \subset \phi_n^{-1}(\Delta)$. Let $\hat{f}_n$ be the restriction to the convex open set $S \cap \phi_n^{-1}(\Delta)$ of $f \circ \phi_n$. Note that $\Delta \cap \phi_n^{-1}(\Delta)$ is not empty since it contains $x'_n$. If follows that for any $m$ and $n$, $f_m$ and $f_n$ coincide near $x'_n$ with $f$, so coincide near $x'_n$, and by affine continuation coincide on $\text{Dom}(f_n) \cap \text{Dom}(f_n)$, the latter set being convex, hence connected. It follows that, given $z \in S$, the value of $\hat{f}_n(z)$ for the various $n$ such that $z \in \text{Dom}(\hat{f}_n)$ is independent of $n$. We deduce that $f$ has an extension $\hat{f}: S \to X^*$ that is an affine immersion (and actually takes values in $f(\Delta)$). The domain of $\hat{f}$ is the convex set $S$ and near $x'_n$, $\hat{f} \circ \phi_n = f \circ \phi_n = f = \hat{f}$, i.e.

$$ \phi_n \in \text{Hol}(\hat{f}). $$

For $n$ big enough, $\phi_n(S) \cap S$ contains $\{x_n + z; z \in S_0 \text{ and } a_{\phi_n}z \in S_0\}$, hence is unbounded. Up to reindexing the sequence, we assume this is the case for all $n$. We now apply Proposition 5.26 to $\hat{f}$. Because $\phi_n \in \text{Hol}(\hat{f})$ and $\phi_n(S) \cap S$ is unbounded, it follows that $\hat{f}$ is not injective on any neighborhood of $\infty$, so we cannot be in Case (i) of Proposition 5.26. If $X^*$ were a full translation cylinder or a translation torus, $\text{Hol}(\hat{f})$ could only contain translations, so these cases are excluded too. Case (iii), of $\hat{f}$ factoring near $\infty$ to an semi-cylinder embedding, is incompatible too with $\phi_n \in \text{Hol}(\hat{f})$. Hence either

(1) $X = X^*$ is an affine torus,

(2) $\hat{f}$ factors near $\infty$ to an embedding through a anti-conical singularity of opening angle $\le \theta$,

(3) $\hat{f}$ factors near $\infty$ to an immersion through a dilation cylinder.

More precisely denote $\theta$ the opening angle of $C$. In cases (2) and (3), Proposition 5.26 tells us that $\hat{f} = g \circ \pi \circ \iota \circ L$ holds on a neighborhood $V$ of $\infty$ in $C$, where:

* $L \in \text{Aut}(C)$ is such that $0 \notin L(S_0)$,

* $\iota$ is an embedding of $L(S_0)$ in $\tilde{C}^*$,

* $\pi : \tilde{C}^* \to \tilde{C}^* / \Lambda$,

* $\Lambda$ is the group generated by the lift to $\tilde{C}^*$, and with rotation angle in $(-\pi, \pi)$, of $z \mapsto a_*z$

* $a_* \in \tilde{C}^*$, $a_* \ne 1$ and $\arg(a_*) \in (-\theta, \theta)$,

* $g$ is an embedding from $\pi \circ \iota \circ L(V)$ to $X^*$.

We denote $\phi_* = L^{-1} \circ (z \mapsto a_\phi z) \circ L \in \text{Aut}(C)$, which fixes $z_1 := L^{-1}(0)$ and has dilation factor $a_*$. Proposition 5.26 tells us that the only elements $\phi \in \text{Hol}(\hat{f})$ such that $|\arg(a_\phi)| < \theta$, must be powers of $\phi_*$. In particular, they all fix $z_1$. Applying this to $\phi_n$, we get $\phi_n(z_1) = z_1$. Then $z_1 - x_n = \phi_n(z_1) - \phi_n(x'_n) = a_{\phi_n} \times (z_1 - x'_n) \to 0$ so $x_n \to z_1$. So $z_1 = z_0$, i.e. every $\phi_n$ fixes $z_0$ and $\phi_*$ fixes $z_0$.

So in case (2), the relation $\hat{f} \circ \phi_* = \hat{f}$ holds on the sector based on the tip $z_0$ of $S_0$ and the form $S_0 \cap \phi_*^{-1}(S_0)$ and $\hat{f}$ factors as $\hat{f} = g \circ \pi$ where $\pi : S_0 \to S_0 / (z \sim \phi_*(z))$ and $\phi_*$ fixes $z_0$ and $\arg(a_{\phi_*}) \in (-\theta, 0) \cup (0, \theta)$ where $\theta$ is the opening angle of $S_0$. The map $g$ is supposed to be injective near $\infty$. However, for all $n$, the relation $f(\phi_n(z)) = f(z)$ holds on an unbounded sector contained in $S_0$. For $n$, then $z$, very big, $|\arg(\phi_n(z)/z)| < \arg(a_{\phi_*})$, so $\phi_n(z)$ and $z$ project to distinct points of the skew cylinders, that are close to $\infty$, but mapped to the same point by $g$, leading to a contradiction.

Hence we are in case (3) and since $z_1 = z_0$, $\phi_*(S_0) = S_0$ and this proves Proposition 5.29.
From our results on triangle immersions and on plane immersions, we deduce the analo-
gous result for immersions of convex sets at flat points of their boundaries, which is crucial
in the case of disks for the construction of the Delaunay decomposition in Section 6.

**Corollary 5.30.** Consider an affine immersion $f$ of an open convex subset $\Delta$ of $\mathbb{C}$ to $X^*$. We assume that $f$ does not extend continuously at $z_0 \in \partial\Delta$ and $\Delta$ has a tangent at $z_0$. Then $f$ extends to an immersion $\hat{f}$ of the infinite half-plane $H$ containing $\Delta$ and delimited by the tangent to $\Delta$ at $z_0$.

Moreover, one of the following statements holds:

(i) $X$ is an affine torus;

(ii) $\hat{f}$, hence $f$, factors to an embedding of a dilation cylinder of angle $\pi$: $\hat{f} = g \circ \pi$ where $\pi: H \to H/\langle\phi\rangle$, $g$ is an affine embedding and $\phi$ is a dilation fixing $z_0$.

In Case (ii) every radial geodesic is mapped to an attracting closed geodesic of that
cylinder.

**Proof.** Proposition 5.29 proves that for any triangle $T$ of vertex $z_0$ contained in $\Delta$, $f|_T$ extends to an infinite sector $\mathcal{C}$ of vertex $z_0$ containing $T$, and this extension actually coincides with $f$ on $\mathcal{C} \cap \Delta$ by the affine identity principle. Using a sequence of triangles whose angle a $z_0$ is increasing, bisected by the normal to $\partial\Delta$ at $z_0$, and tends to $\pi$, the extensions extend each other by the affine identity principle, and their union defines an extension $\hat{f}$ of $f$ to the half-plane $H$ containing $\Delta$ whose boundary line is tangent to $\partial\Delta$ at $z_0$. The extension does not extend continuously at $z_0$, for $f$ would, and the end of the statement of Proposition 5.23 proves that either $X$ is an affine torus, or $\hat{f}$, hence $f$, factors through the embedding into $X^*$ of a dilation cylinder of angle $\pi$. The fixed point of $\phi$ must be $z_0$, for otherwise there would be continuous extension at $z_0$. $\square$

**5.9. Boundary of immersed disks and cylinders.** We draw the following summary of
possibilities in the specific case of a disk, in view of the previous results.

**Lemma 5.31.** Given an affine immersion of a disk $\Delta$ to $X^*$, either

(i) there is a point $z_0 \in \partial\Delta$ where $f$ does not extend continuously; then $f$ has a continuous extension on $\partial\Delta \setminus \{z_0\}$, which maps in $X^*$; we are in the situation of Corollary 5.30, i.e. either $X$ is an affine torus or the image of $f$ is an embedded Reeb cylinder of angle $\pi$;

(ii) $f$ has a continuous extension to $\partial\Delta$, taking values in $X$; only finitely many points map to $S$.

**Proof.** If there is a point of the boundary where $f$ does not extend continuously then Corollary 5.30 shows we are in Case (i). Otherwise, $f$ has a continuous extension $\bar{f}$ to $\partial\Delta$. Since $\bar{f}^{-1}(S)$ is closed and discrete by Lemma 5.9 in the compact set $\partial\Delta$, it follows that it is finite. $\square$

This also applies when $\Delta$ is any bounded convex subset of $\mathbb{C}$ with a tangent at every
point of its boundary, i.e. without angular points.

It has been proved that every (possibly self-intersecting) closed geodesic embeds in a
continuous family of closed geodesics forming an immersed cylinder. Now, we can describe
the boundary of a cylinder.

**Corollary 5.32.** Given an affine immersion $f$ of a translation or dilation cylinder $C$ to $xX^*$, $f$ extends continuously to the boundary of the cylinder. The image of a boundary component is either a (possibly self-intersecting) closed geodesic or the union of finitely many (possibly intersecting and self-intersecting) saddle connections joining apexes of $S$.

Moreover, if $f$ is an embedding, then the closed geodesics or individual saddle connec-
tions in the image of the boundary component have no self-intersection (but two saddle
connections may have non-empty intersection, as in the example below).
**Example 5.33.** As an example, consider the cushion, a surface formed from two copies of a square (one with inverted orientation) glued along their facing boundaries. It has 4 singularities, which are Fuchsian conical. An embedded cylinder is obtained by removing two opposite edges of the original square, which are particular saddle connections. Each boundary circle is mapped in a 2 : 1 way to one of these removed edges.

Proof of Corollary 5.32. Let $z_0 \in \partial C$. Assume by contradiction that $f$ does not have a continuous extension at $z_0$. We defined $\partial C$ as a closed subset of some affine surface $Y$ containing $C$. Consider an embedded (injective) disk $D$ centered on $z_0$ in $Y$. The set $D \cap C$ can be identified with a half-disk, which we call $\Delta$, in $\mathbb{C}$. In our hypothetical situation, the restriction of $f$ to $\Delta$ is not continuous. Corollary 5.30 then implies that $f|_{\Delta}$ extends to an immersion $\hat{f}$ of the half-plane $H$ delimited by the line through $z_0$ supporting the straight part of $\partial\Delta$ and satisfying $\hat{f} \circ \phi = \hat{f}$ where $\phi \in \text{Aut}(\mathbb{C})$ is a dilation of ratio $s \neq 1$ and fixing $z_0$. Following Corollary 5.22, all the elements of $\text{Hol}(\hat{f}|_H)$ commute and therefore fix $z_0$. But $H$ contains the universal covering $\tilde{C}$ of $C$, on which $f$ lifts to a map $\tilde{f}$ which has an element $\phi \in \text{Aut}(\mathbb{C})$ in its holonomy set that does not fix $z_0$. By affine continuation, $\tilde{f}$ must coincide with $\hat{f}$ on $\tilde{C}$, and $\text{Hol}(\tilde{f})$ contains $\phi$ which thus should fix $z_0$, leading to a contradiction.

Lemma 5.3 implies many things:

* That the points of the component $B$ of $\partial C$ containing $z_0$ and mapped to a singularity form a discrete subset $A$. It is also a closed subset, being the preimage of the closed subset $S$ of $X$ by the continuous extension. Hence $A$ is finite.

* That the singular points must be conical Fuchsian singularities or irregular singularities.

* That near every other point, there is an affine extension. In particular the boundary between two points of $A$ map by $f$ to the support of a geodesic of finite time span starting and ending at singularities, i.e. a saddle connection. If $A$ is empty then $B$ is mapped to the support of a closed geodesic.

Finally, assume that if $f$ is not injective on a component of $B \setminus A$, meaning that two boundary points $z_0, z_1$ mapped to $X^*$ have same image under $f$. Since this component correspond to a closed geodesic or a geodesic arc, $z_1$ and $z_0$ are mapped to a self-intersection point of the geodesic, and at such a point the directions of the geodesic cannot be opposite. It follows that we can find points in the interior of $C$ that have same image under $f$. Thus, if $f$ is an embedding, its boundary geodesics and individual saddle connections have no self-intersection. $\square$

## 6. DELAUNAY DECOMPOSITION

In this section, we organize the affine immersions of planes, half-planes, and disks into a finite type affine surface into the *Delaunay category*, introduced in Section 6.1, whose structure will allow us to describe the qualitative geometry of the affine surface. We define maximal elements of the Delaunay category and prove that each element of the Delaunay category is the restriction of such an immersion.

In Section 6.2, we characterize the maximal elements of the Delaunay category of a given affine surface as affine immersions which exhibit a certain behavior on the boundary.

In Section 6.3, we define a deformation process of maximal elements, which we call pivoting. This allows to give a graph structure to the set of maximal elements, which we call the Delaunay spine, and that can be naturally embedded in the affine surface, blown up at some points.

In Section 6.4 we define Delaunay segments using one of the maximal immersion types: type A (disk, flexible). We prove that they form saddle connections that are embedded
(not self-crossing) and that they do not cross each other. We also prove, using the counting
results in Section 4.1, that there are only finitely many Delaunay segments.

In particular, in Section 6.6, we classify the affine surfaces without Delaunay segments,
referred to as the exceptional affine surfaces (proving thus Theorem 1.6).

In Section 6.5, we define open Delaunay polygons, as image by rigid maximal disk immersions of the interior convex hull of points mapped to singularities and prove that they are embedded and contain no point on a Delaunay segment. We also define the exterior of $X$ as the union of images of all half-plane immersions (plus the non-conical Fuchsian singularities), and the core as its complement. We prove that the core is the union of the open Delaunay polygon interiors, the Delaunay segment interiors, the conical Fuchsian singularities and the irregular singularities.

In Section 6.7, we describe the connected components in the Delaunay decomposition of non-exceptional surface (proving Theorems 1.7 and 1.10).

## 6.1. Delaunay category.
Affine immersions of standard open domains of the complex plane into a finite type affine surface assemble into an ordered structure that we present below.

We recall that affine *immersions* are just affine maps, i.e. maps that are locally affine in charts. We emphasize that they do not need to be injective (they only have to be locally injective). We call affine *embedding* the injective immersions. For brevity, we will often abbreviate affine immersion as immersion, and affine embedding as embedding.

**Definition 6.1.** For a finite type affine surface $(X, \nabla, S)$, the *Delaunay category* $\mathcal{D}el$ is defined as follows:

* objects of $\mathcal{D}el$ are pairs $(\Delta, f)$ where $\Delta$ is an open disk in the canonical affine plane $\mathbb{C}$, an open half-plane in $\mathbb{C}$ or the whole plane, and $f$ is an affine immersion from $\Delta$ to $X^*$;

* an arrow between two pairs $(\Delta_0, f_0)$ and $(\Delta_1, f_1)$ is an affine embedding $\phi : \Delta_0 \to \Delta_1$ such that $f_0 = f_1 \circ \phi$.

The existence of an arrow between two objects of the Delaunay category is a preorder relation: it is reflexive and transitive. It is not antisymmetric for two reasons. First, for any automorphism $\phi$ of $\mathbb{C}$, the element $(\phi^{-1}(\Delta), f \circ \phi)$ is isomorphic to $(\Delta, f)$ in this category. Second, in the case $X^*$ is a full cylinder $\mathbb{C}/\mathbb{Z}$, for any two half-planes $H$ and $H'$ with parallel non-horizontal boundary lines, there is an arrow from $\pi|_H$ to $\pi|_{H'}$ and an arrow from $\pi|_{H'}$ to $\pi|_H$: assuming $H$ contains a neighborhood of $-\infty$ in $\mathbb{R}$, and that $H' = T_x(H)$ with $x \in \mathbb{R}$, then for any $n \in \mathbb{Z}$ such that $x+n \le 0$, we have $\pi'_{H'} = \pi_H \circ T_n|_{H'}$, while for any $n \in \mathbb{Z}$ such that $n \le x$ then $\pi_H = \pi'_{H'} \circ T_n|_H$.

The notion of *maximal elements* (up to isomorphism) for this relation support important geometric information on the complex affine structure.

**Definition 6.2.** For a finite type affine surface $(X, \nabla, S)$, a *maximal element* of the Delaunay category is a pair $(\Delta, f)$ such that for any $(\Delta_1, f_1)$ for which there is an arrow $\phi : (\Delta, f) \to (\Delta_1, f_1)$ then $\phi^{-1}$ is an arrow $(\Delta_1, f_1) \to (\Delta, f)$ (i.e. $\phi(\Delta) = \Delta_1$).

**Remark 6.3.** In the more general setting of complex projective structures (also called Möbius geometry), we do not distinguish between disks and half-planes. The space of immersed disks in a complex projective surface is locally modeled on the space of oriented circles in $\mathbb{CP}^1$, which identifies with a three-dimensional de Sitter space. The relation defined by the existence of an arrow between two immersions then corresponds to the causal structure induced by the Lorentzian metric, see [Sca99] for details.

**Lemma 6.4.** An element $(\Delta, f)$ of the Delaunay category is maximal if and only if there is no affine extension of $f$ to a disk, half-plane or whole plane, containing $\Delta$.
*Proof.* If $(\Delta, f)$ has an extension $(\Delta_1, f_1)$ with $\Delta \subsetneq \Delta_1$, then the injection $\phi(z) = z$ from $\Delta$ to $\Delta_1$ is an arrow from the first to the second but $\phi(\Delta) \neq \Delta_1$, so $(\Delta, f)$ is not maximal.

Conversely if $(\Delta, f)$ has no such extension and if there is some arrow $\phi : (\Delta, f) \to (\Delta_1, f_1)$ then $\phi$ is the restriction of an element $\hat{\phi} \in \mathrm{Aut}(\mathbb{C})$ and $f_1 \circ \hat{\phi}$ provides an extension of $f$, so $\hat{\phi}^{-1}(\Delta_1) = \Delta$ and $\phi^{-1}$ is an arrow from $(\Delta_1, f_1)$ to $(\Delta, f)$. $\square$

We will characterize maximal elements in Section 6.2. We first state and prove below the existence of maximal elements above a given one.

**Lemma 6.5.** Consider a finite-type affine surface $(X, \nabla, S)$. Then every element $(\Delta, f)$ of the Delaunay category is the restriction of a maximal element.

*Proof.* In view of Lemma 6.4, it is enough to prove that the set of open disks/half-planes/whole planes in $\mathbb{C}$ containing $\Delta$ and on which $f$ has an affine extension mapping in $X^*$, has a maximal element for the inclusion.

If $\Delta$ is a half-plane, consider the set of all open half-planes containing $\Delta$ and on which $f$ has an affine extension to $X^*$. Consider their union: it is either the whole plane $\mathbb{C}$ or a half-plane $\hat{H}$. In any case, for any two open half-planes $H_1, H_2$ containing the half-plane $\Delta$, one must contain the other. If $f$ has affine extensions to $H_1, H_2$, then they must coincide on the smallest of $H_1$ or $H_2$, by the affine identity theorem since they coincide on $\Delta$. It follows that we can define $\hat{f}(z)$ for $z \in \mathbb{C}$ or $\hat{H}$ as the union of all these extensions, i.e. as $f_1(z)$ for any extension $f_1$ defined on a half-plane $H_1$ containing $z$.

If $\Delta$ is a disk, consider the supremum $R \in (0, +\infty]$ of the radii of disks $\Delta'$ containing $\Delta$ and on which $f$ can extend. There exists a sequence $\Delta_n$ whose radii sequence $R_n$ tends to $R$. If $R$ is finite then by extraction we may assume that their centers converge too, to some $c \in \mathbb{C}$. As in the case for planes, we can then use the affine identity theorem to define an extension $\tilde{f}$ on $B(c, R)$ using an exhaustion by connected compact subsets containing a closed disk contained in $\Delta$. If $R = +\infty$ then either the distance from the center $z_1$ of $\Delta$ to the circle $\partial\Delta'$ tends to infinity, in which case we can extract a nested subsequence and $f$ extends to $\mathbb{C}$. Or this distance stays bounded and we can extract a subsequence such that the $\Delta_n$ tend to a half-plane $H$ and we can proceed as in the case $R$ is finite to get an extension to $H$ by exhaustion. It may happen that $H$ is not maximal. But we know that the extension to $H$ has a maximal extension because we treated the case $\Delta$ is a half-plane. $\square$

## 6.2. Maximal elements in the Delaunay category.
In the following two statements, we prove that maximal immersion have specific boundary behavior.

But first let us recall what we proved concerning the boundary behavior of general immersions $f$ of disk and half-planes in finite type affine surfaces. By Lemma 5.31 and the end of Proposition 5.23, disk and half-plane affine immersions continuously extend to the whole boundary except maybe one point (in which case $f$ factors through a Reeb cylinder embedding). Denote $\bar{f}$ the continuous extension. Denote

$$A = \bar{f}^{-1}(S)$$

the set of points (on the boundary) that are mapped to a singularity by the extension. For half-planes, using Lemma 5.9 and the classification in Proposition 5.23, we get: if there is a discontinuity point $z_0$, then $A$ is a finite union of orbits under a common dilation fixing $z_0$; if there is no discontinuity point, either $A$ is finite or $f$ factors through a cylinder embedding and $A$ a finite union of orbits under a common translation of the boundary line. For disks, by Lemma 5.31, in the presence of a discontinuity then $A$ is empty, while if there is no discontinuity then $A$ is finite.

The results of the present section will in particular prove that, given a finite type affine surface $(X, \nabla, S)$, an element $(\Delta, f)$ of the Delaunay category is maximal if and only if it is one of the following:
- the immersion of a disk with at least two points in A;

- the immersion of a half-plane with at least one point in A;

- the immersion of a half-plane that does not extend at some point of the boundary;

- the immersion of the whole plane.

**Lemma 6.6.** Consider a finite type affine surface $(X, \nabla, S)$ and an affine immersion $f: \Delta \to X^*$ of a disk. If $(\Delta, f)$ is maximal then $f$ has no discontinuity point on $\partial\Delta$ and $\bar{f}$ maps at least two (boundary) points to singularities. Conversely, an affine immersion of a disk such that $\bar{f}$ sends at least two points to singularities is maximal (hence has no discontinuity point).

*Proof.* Assume $(\Delta, f)$ is maximal. In the presence of a discontinuity, we saw $f$ would factor to a Reeb cylinder embedding, which implies that $f$ extends to a half-plane containing the disk, so $f$ would be non-maximal. So there is no discontinuity: $f$ extends continuously to $\partial\Delta$.

If the whole boundary maps to regular points, there is a local affine extension near every boundary point, and using a compactness argument and the affine identity theorem, we can extend to a disk of the same center but bigger radius.

If there is only one point $z_0 \in \partial\Delta$ mapped to a singularity, then a similar procedure can be applied to extend to a bigger disk tangent to $\Delta$ at $z_0$, using at $z_0$ that we have an extension to a slit neighborhood (Lemma 4.10). See Figure 37.

So there are at least two points on $\partial\Delta$ that are mapped to singularities by the continuous extension of $f$.

Conversely, if two boundary points map to singularities, any extension of $f$ must be defined on a set avoiding these two points. But there is no set among open disks, open half-planes or the whole plane, that strictly contains $\Delta$ but does not contain any of the two points. $\square$

<figure><img src="image_39.png" /><figcaption>FIGURE 37. A disk immersion that has an affine extension near every point of its boundary except one where it has a slit extension, can be extended to a bigger tangent disk.</figcaption></figure>

**Lemma 6.7.** Consider a finite type affine surface $(X, \nabla, S)$ and affine immersion $f: H \to X^*$ of an open half-plane. Then $(H, f)$ is maximal if and only if either it has a discontinuity point on $\partial H$, or its extension $\bar{f}$ to $\partial H$ maps a point$^{22}$ to a singularity (these two cases are not exclusive).

*Proof.* Discontinuities and singularities obviously prevent extension.

For the converse, we use Proposition 5.23. By hypothesis, $f$ may satisfy any case of that statement.

<footnote>$^{22}\partial H$ denotes the boundary of $H$ in $\mathbb{C}$, i.e. the point at infinity is excluded.</footnote>
In Cases (i) ($f$ is an embedding) or (iv) ($f$ factors to an embedding of an anti-conical Fuchsian singularity standard neighborhood, of opening angle $< \pi$), Proposition 5.23 tells us in particular that, as $z$ tends to $\infty$, $f(z)$ tends to an anti-conical singularity or a swath of an irregular singularity. Applying Lemma 4.11 to some half-lines contained in $H$ that are parallel to $\partial H$, we show that $f$ extends on a sector $S$ of angle strictly larger than $\pi$ that contains all the boundary line except possibly a segment $I$ of finite length. Points mapped to a singularity are necessarily in $I$. If there were no such point, we may combine the extension of $f$ to sector $S$ and a finite union of disks covering $I$ as in the disk case above, and deduce that $f$ extends to an open half-plane $H'$ containing $H$ as a strict subset.

In Case (ii), half-plane immersion are very explicit: for full cylinders and translation tori, they always extend to a bigger half-plane, for affine tori, this is the case too if the lift of the half-plane in $\tilde{C}^*$ has a boundary that does not contain 0, otherwise continuous extension fails at this point.

In Case (iii), $f$ factors through the embedding of a semi-infinite translation cylinder. The latter bounded by a chain of finitely many saddle connections (see Corollary 5.32) or by a periodic geodesic. In the second case, $f$ would extend to a bigger half-plane (by a similar argument as for disks above), so we are in the first case.

Case (v) is when $f$ factors through the embedding of a Reeb cylinder of angle $\pi$, more precisely there exists a non-trivial dilation $\phi \in \text{Aut}(\tilde{C})$ whose fixed point does not belong to $H$, and a factorization $f = g \circ \pi|_H$ where $\pi: H' \to H'/(\langle\phi\rangle)$ is the quotient map to a Reeb cylinder of angle $\pi$, $H'$ is the half-plane containing $H$ and whose boundary passes through the fixed point of $\phi$ and $g$ is an affine embedding from the Reeb cylinder into $X^*$. Since $g \circ \pi$ defines an extension of $f$ to $H'$, we actually have $H' = H$. So there is a discontinuity on $\partial H$. $\square$

As a consequence:

**Proposition 6.8.** Consider a finite type affine surface $(X, \nabla, S)$ that is not an affine torus. Let $(H, f)$ be a maximal element of the Delaunay category with $H$ a half-plane. Then $X$ does not admit an immersion of the whole plane. Moreover $(H, f)$ satisfies one of these three mutually exclusive statements:

(i) $f$ tends at infinity to an anti-conical singularity or a irregular singularity swath, it extends continuously at every point of the boundary line, the number of points mapped to points of $S$ is finite and at least one, it maps the segment between two consecutive such points to a saddle connection.

(ii) $f$ is the universal covering of a semi-infinite translation cylinder, it extends continuously at every point of the boundary line, the boundary line is mapped to a chain of finitely many saddle connections;

(iii) $f$ is the universal covering of an embedded Reeb cylinder of angle $\pi$, it extends continuously at every point of the boundary line excepted exactly one point, the image of the boundary is the image of the two boundary circles of the cylinder, each of which being a closed geodesic or a finite union of saddle connections;

*Proof.* We cannot be in Case (ii) of Proposition 5.23: we excluded affine tori and for full cylinders and translation tori, the only maximal immersions are that of the whole plane.

Then, the only case for which a discontinuity can arise on $\partial H$ is, by the end of Proposition 5.23, Case (v) of that statement, when the fixed point of the generator $\phi$ is on $\partial H$. With this and Corollary 5.32, we get Case (iii) of the present statement.

Otherwise there is continuous extension to $\partial H$ and we are not in Case (v) of Proposition 5.23 for otherwise we would be able to extend $f$, as already noted in the proof of Lemma 6.7.

In Cases (i) or (iv) of Proposition 5.23, we saw that $f$ extends near $\infty$ to a sector containing the boundary line and tends to an anti-conical singularity or an irregular singularity
swath. By Lemma 5.9 there is a finite number of point on $\partial H$ mapped to singularities. By Lemma 6.7 there is at least one. We get Case (i) of the present statement.

In Case (iii) of Proposition 5.23, we can apply Corollary 5.32 and lemma 6.7 and get Case (ii) of the present statement. $\square$

We distinguish between *flexible* and *rigid* immersions among the maximal immersions of the Delaunay category of a finite type affine surface.

**Definition 6.9.** A maximal element of the Delaunay category of a finite type affine surface is said to be *flexible* in the following cases:

Type A: (disk, flexible) the immersion of an open disk that extends continuously to its boundary circle where exactly two points are mapped to points of $S$;

Type B: (half-plane, non-cylinder, flexible) the immersion of an open half-plane that extends continuously to its boundary line where exactly one point is mapped to a point of $S$;

Type C: (Reeb, flexible) the immersion of an open half-plane that extends continuously to its boundary line excepted in one point and where no boundary point is mapped to a point of $S$.

We recall that an immersion of a disk or half-plane always extends continuously to all but maybe one point of the boundary and that the set of points mapping to a singularity is discrete (all points are isolated).

**Definition 6.10.** A maximal element of the Delaunay category of a finite type affine surface is said to be *rigid* in the following cases:

Type I: (disk, rigid) the immersion of an open disk that extends continuously to its boundary circle where at least three of points are mapped to points of $S$;

Type II: (half-plane, non-cylinder, rigid) the immersion of an open half-plane that extends continuously to its boundary line where only a finite number of points, at least two, are mapped to points of $S$;

Type III: (semi-cylinder, rigid) the immersion of an open half-plane that extends continuously to its boundary line where an infinite and translation-periodic subset of points are mapped to points of $S$;

Type IV: (Reeb, rigid, $> \pi$) the immersion of an open half-plane that extends continuously to its boundary line excepted in one point $z_0$ and where the set of boundary points mapped to points of $S$ is non-empty, invariant by non-trivial dilation fixing $z_0$, and contained on one side of $z_0$;

Type V: (Reeb, rigid, $\pi$) same as Type IV but with points mapped to $S$ on both sides of $z_0$;

Type VI: (whole plane) the immersion of a whole plane.

Using the previous results, we check immediately that, for finite type affine surface, every maximal element of the Delaunay category is either rigid or flexible.

**6.3. Affine Delaunay spine and pivoting.** Given a maximal element $(\Delta, f)$ of the Delaunay category where $\Delta$ is not the whole plane, we define a way to deform it which we call *pivoting*. Consider the boundary $\partial\Delta$ from which we remove the discontinuity point if there is one, and every point mapped to $S$ by the continuous extension (then by Lemma 5.9 these singularities are conical Fuchsian or irregular). One gets an open subset of $\partial\Delta$, which has at least two connected components, possibly infinitely many if $\Delta$ is a half-plane. Choose one component, denote it $J$ and call it the *egress component*. It is an open arc of circle, an open segment, or an open half-line.

Case 1: the component $J$ is bounded. Then $J$ is bounded by two continuity points $A$ and $B$ mapped to singularities. There is thus an extension of $f$ (see Lemma 4.10) to a set $V$ containing $\Delta$, a neighborhood of the open arc/segment $J$ and two slit neighborhoods of
A and B, and we can choose the slits tangent to $\partial\Delta$ at these points. As a consequence, consider pencil of circles/line passing through A and B, which form a continuous family we can parametrize by the angle $\theta \in \mathbb{R}/\pi\mathbb{Z}$ they make at A with $\partial\Delta$, with $\theta$ small and positive corresponding to the existence of an arc outside $\Delta$ but close to J. Call $C_\theta$ the circle/line of parameter $\theta$ in the pencil. Then there is some $\epsilon > 0$ such that for all $\theta \in [0, \epsilon]$, V contains the disk $\Delta'$ bounded by the circle of parameter $\theta$. By Lemma 6.6, $(\Delta', \hat{f}|_{\Delta'})$ is a maximal element of the Delaunay category.

Case 2: the component $J$ is unbounded, i.e. it is a half-line. If it is bounded by a discontinuity point $z_0$ then $f \circ \phi = f$ for some non-trivial dilation $\phi$ fixing $z_0$. Let $[a, b) \in J$ be a fundamental domain of the action of $\phi$ on $J$. Every point of $J$ has a neighborhood on which $f$ extends, and by compactness of $[a, b]$ and $\phi$-invariance we can extend $f$ to an affine immersion defined on the union of $H$ and of an open infinite sector bisected by $J$. This immersion still satisfies $\hat{f} \circ \phi = \hat{f}$. If $J$ is bounded by a point $z_0$ mapped to a singularity, then by Lemma 4.10 near $z_0$ and regular points, and Proposition 5.23 near infinity (we are in Cases (i) or (iv) of that statement), we can also affinely extend $f$ to the union of $H$ and of an open infinite sector bisected by $J$ and the extension extends continuously to the boundary of its domain. In both cases the domain of $\hat{f}$ contains every half-plane $\Delta'$ containing $J$ and obtained by rotation $H$ around $z_0$ by a small enough amount. the half-plane bounded by the rotated line and containing $J$. Then $(\Delta', \hat{f}|_{\Delta'})$ is still a maximal element of the Delaunay category.

See Figure 38 for an illustration of these deformations.

<figure><img src="image_40.png" /><figcaption>FIGURE 38. Pivoting maximal disk/half-plane immersions. The egress component $J$ is in blue.</figcaption></figure>

If $J$ is a bounded segment then: If $\Delta$ is a plane and $J$ is a bounded segment, then for all $\theta \in (0, \pi)$, $\Delta_\theta$ is a disk and we define $\theta_0 = \pi$ and $\Delta_{\theta_0}$ as the open half-plane bounded by $\partial\Delta$ but opposite to $\Delta$. If $\Delta$ is a disk, then there is some $\theta = \theta_0$ for which $C_\theta$ is the line through A and B. We define $\Delta_{\theta_0}$ as the half-plane bounded by $C_{\theta_0}$ and which contains $J$. Consider then the supremum $\theta_1 \in (0, \theta_0]$ of $\theta \in (0, \theta_0)$ such that $f$ extends to an
affine immersion of $\Delta \cup \Delta_{\theta}$. Then $f$ extends to an immersion $\hat{f} : \Delta_{\theta_1} \cup J \cup \Delta \to X^*$ and $(\Delta_{\theta_1}, \hat{f}|_{\Delta_{\theta_1}})$ is a rigid maximal element of the Delaunay category.

If $\Delta$ is a plane and $J$ is an infinite half-line, then for all $\theta \in (0, +\infty)$, $\Delta_{\theta}$ is a half-plane. There is a maximal $\theta_1 \in (0, +\infty]$ and for all $\theta \in I := [0, \theta_1] \cap \mathbb{R}$ an immersion $f_{\theta} : \Delta_{\theta} \to X^*$ such that for all $\theta, \theta' \in I$ with $|\theta - \theta'| < 2\pi$, $f_{\theta}$ and $f_{\theta'}$ coincide on $\Delta_{\theta} \cap \Delta_{\theta'}$.

We get a family of maximal elements of the Delaunay category parametrized by $\theta \in [0, \theta_1] \cap \mathbb{R}$. We call this process *pivoting*.

*The affine Delaunay spine.*

This process naturally endows the set of isomorphism classes of maximal elements of the Delaunay category with a graph structure. The vertices are the classes of rigid element, and the edge interiors points are the classes of flexible elements: indeed, flexible element have exactly two egress components, while rigid ones have at least three (with the exception of whole plane immersions, in which case the graph is a single point). It also comes with a natural metric along edges given by the pivoting angle. In this case there may be infinitely long edges in the presence of semi-infinite Reeb cylinders (i.e. of a Reeb type Fuchsian singularity) or irregular singularities. This graph is an analogue of the Voronoi graph of a finite set of marked points in the Euclidean plane. Actually in that setting, the Delaunay decomposition we defined coincides with the classical one, and it is known to be dual to the Voronoi decomposition: the center of all maximal disks trace the edges and vertices of the Voronoi decomposition associated to the marked point. In our setting of a finite type affine surface, the center of maximal disks also trace a graph on $X^*$, which can be called the Voronoi decomposition, and is homeomorphic to the part of the Delaunay spine excluding the half-plane immersions. This is interesting because distance is not well-defined on affine surfaces (this is already the case for dilation surfaces, see [BGT25]). To embed the full Delaunay spine, one would have to blow up non-conical singularities and irregular singularities in a certain way.

**6.4. Delaunay segments.** We recall here the definition of a Delaunay segment introduced in Definition 1.4. Immersions of type A are immersions of open disks that extend continuously to the boundary circle, with exactly two points on the boundary mapped to singularities of $\nabla$. Delaunay segments are the images of the chords drawn between the two singular points under these immersions. In this section, we show that Delaunay segments are non-self-intersecting saddle connections with mutually disjoint interiors. These claims follow from a lemma of Euclidean geometry, which state without proof.

**Lemma 6.11.** Let $D_1, D_2$ be two open disks in the plane with a nonempty intersection $D_1 \cap D_2$ and such that neither $D_1 \subset D_2$ nor $D_2 \subset D_1$. We denote by $I, J$ the intersection points of their boundary circles $\partial D_1$ and $\partial D_2$. Let $\alpha_1$ (resp. $\alpha_2$) be the circular arc drawn between $I, J$ contained in $\partial D_1$ and disjoint from $D_2$ (resp. contained in $\partial D_2$ and disjoint from $D_1$). Consider two open segments $(A_1, B_1)$, $(A_2, B_2)$ with endpoints respectively in $\alpha_1$ and $\alpha_2$, see Figure 39. If $(A_1, B_1)$ and $(A_2, B_2)$ intersect, then they both coincide with $(I, J)$.

**Corollary 6.12.** In any affine surface $(X, \nabla, S)$, the interior of Delaunay segments are embedded finite straight open segments drawn between points of $S$ (they have no self-intersection outside the endpoints).

*Proof.* Consider a Delaunay open segment $e$ of $(X, \nabla, S)$ that is the image of a chord $(A, B)$ by the affine immersion of an open disk $\Delta$. We assume by contradiction that $e$ admits a self-intersection in some regular point $z \in X^*$ and consider two distinct points $x_1, x_2$ in $(A, B) \cap f^{-1}(\{z\})$.

By Section 5.5, there exists $g \in \mathrm{Aut}(\mathbb{C})$ such that $g(x_1) = x_2$ and $f = f \circ g^{-1}$ holds on $\Delta \cap g(\Delta)$. This allows to extend $f$ to an immersion $\hat{f} : \Delta \cup g(\Delta) \to X^*$. Both
<figure><img src="image_41.png" /><figcaption>FIGURE 39. Illustration of Lemma 6.11</figcaption></figure>

immersions $\hat{f}|_{\Delta} = f$ and $\hat{f}|_{g(\Delta)} = f \circ g^{-1}$ are immersions of type A (exactly two points of their boundary circles are mapped to singularities of the affine structure; for the second these are $g(A)$ and $g(B)$). It follows that neither $\Delta \subset g(\Delta)$ nor $g(\Delta) \subset \Delta$ can hold: indeed if we had, say, $g(\Delta) \subset \Delta$ then since neither $g(A)$ nor $g(B)$ can belong to $\Delta$, this means $g(\Delta) = \Delta$. In that case, the points mapped to $S$ by $\bar{f} \circ g^{-1}$ are only $A$ and $B$, so $\{g(A), g(B)\} = \{A, B\}$. So either $g = \text{id}_C$, which contradicts $g(x_1) = x_2$, or $g$ permutes $A$ and $B$, which means $[A, B]$ is a diameter of $\Delta$ but then $g$ fixes the midpoint of $[A, B]$ in $\Delta$, which is impossible by Lemma 5.20.

By definition, the intersection $\Delta \cap g(\Delta)$ is nonempty because it contains a neighborhood of $x_2$. Then the open chord $(A, B)$ of $\Delta$ and the open chord $g((A, B))$ of $g(\Delta)$ meet (at $x_2$). On the other hand, their endpoints cannot belong to $\Delta \cup g(\Delta)$ for then they would map by $\hat{f}$ to $X^*$ but also to $S$, a contradiction. They thus form a configuration of segments in $\Delta \cup g(\Delta)$ that is forbidden by Lemma 6.11. $\square$

Similar arguments allow us to show below that Delaunay segments do not intersect each other.

**Corollary 6.13.** In any affine surface $(X, \nabla, S)$, Delaunay segments have disjoint interiors.

*Proof.* We assume by contradiction that two Delaunay segments $e_1, e_2$ of $(X, \nabla, S)$ intersect transversely in some common interior point $z \in X^*$. Then, there is an affine immersion $f$ of a open disk $D_0 \subset \mathbb{C}$ into $X^*$ such that $z \in f(D_0)$.

Then $f^{-1}(e_1)$ and $f^{-1}(e_2)$ form two chords of $D_0$ we extend $f$ along two disks $D_1$ and $D_2$ in such a way that $f|_{D_1}$ and $f|_{D_2}$ are immersions of type A. Observe that neither $D_1 \subset D_2$ nor $D_2 \subset D_1$ are possible since the boundary circle of each of them contains points mapped to singularities. The intersection $D_1 \cap D_2$ is nonempty because it contains a neighborhood of $z$. It follows that $f^{-1}(e_1)$ and $f^{-1}(e_2)$ form a configuration of segments in $D_1 \cup D_2$ that is forbidden by Lemma 6.11. $\square$

Delaunay segments form a system of disjoint arcs drawn between the singularities of the meromorphic connection.

**Lemma 6.14.** There are finitely many Delaunay segments.

*Proof.* This will follow from Lemma 4.5 once we have checked that any non-empty and finite subset $\mathcal{A}$ of the set of Delaunay segments forms an apical arc system (Definition 4.3) and that in the complement in $X$ of the support of $\mathcal{A}$, every ordinary face (topological disk not containing an interior singularity or swath neighborhood) has at least three sides. If such a face $F$ had less, consider a developing map $\phi: F \to \mathbb{C}$. The map $\phi$ has a continuous extension to the closure of $F$. This is immediate near interior points of the Delaunay segments bounding $F$, since they are regular points. Near a conical point, this comes from
the local model of such point. Near an irregular point $p$, this comes from the local model of $p$ and from the fact that $F$ contains no swath neighborhood of $p$. The image $\phi(F)$ is thus a bounded open subset of $\mathbb{C}$, whose boundary is contained in the union of one or two segments. Such a set does not exist. $\square$

**Lemma 6.15.** *Any saddle connection image of a boundary segment by a continuous extension of a half-plane immersion, is a Delaunay segment.*

*Proof.* Denote $(\Delta, f)$ the half-plane immersion, and $[A, B]$ the boundary segment. It is possible to apply the pivot process of Section 6.3 to $(\Delta, f)$ with egress component $(A, B)$. We get a type A Delaunay element whose chord $[A, B]$ defines a Delaunay segment according to Definition 1.4. $\square$

## 6.5. Delaunay polygons and the core.

**Definition 6.16.** An open Delaunay polygon is the image by a rigid maximal immersion $(\Delta, f)$ of type I (disk, rigid) of the interior of the convex hull $P$ of $\bar{f}^{-1}(S)$ (the points of $\partial\Delta$ that map to singular points by the extension of $f$ to $\partial\Delta$).

**Lemma 6.17.** *In Definition 6.16, the image by $\bar{f}$ of the sides of $P$ are Delaunays segments. No diagonal of the convex polygon $P$ is mapped a Delaunay segment.*

*Proof.* For the first claim, note that we can always pivot (see Section 6.3) the immersion around the two endpoints of a side of $P$ so as to get a type A (disk, flexible) immersion.

We proved in Section 6.4 that Delaunay segments do not self-intersect and that no two Delaunay segments can cross either. The only possibility for the presence of a point in a Delaunay segment in an open Delaunay polygon would be that $P$ has at least 4 sides and that a diagonal of $P$ maps to a Delaunay segment. There would be a type A (disk, flexible) immersion $(\Delta', f')$ for which only two boundary points $A', B'$ map to $S$ and such that $f'((A, B)) = f(D)$. Then by composing with an appropriate automorphism of $\mathbb{C}$ and possibly permuting $A'$ and $B'$ we may assume that $A' = A, B' = B$ and $f' = f$ on a neighborhood of $(A, B)$. The points $A$ and $B$ are thus on $\partial\Delta'$. Then $\Delta' = \Delta$ for otherwise $\Delta'$ would contain a point $C \in f^{-1}(S)$, and by affine continuation and continuity we would get $f'(C) \in S$ which is forbidden for affine immersions. From $\Delta' = \Delta$ we get that $f' = f$ is not a type A immersion, but $f$ is of type I: this is a contradiction. $\square$

The next result tells that the interior of Delaunay polygons are always embedded.

**Corollary 6.18.** Given a finite type affine surface $(X, \nabla, S)$, we consider an immersion $f: \Delta \to X^*$ of an open disk into $X^*$ that extends continuously to its boundary circle, where a finite number $m \ge 3$ of boundary points $A_1, \dots, A_m$ are mapped to points of $S$. Then, the restriction of $f$ to the interior $\mathcal{P}$ of the convex hull of $\{A_1, \dots, A_m\}$ is an embedding.

*Proof.* We consider such an immersion and assume by contradiction that $f|_{\mathcal{P}}$ is not injective. Then there exists $\phi \in \text{Aut}(\mathbb{C})$ such that:

* $\phi \neq \text{id}_{\mathbb{C}}$;

* $\phi^{-1}(\mathcal{P}) \cap \mathcal{P} \neq \emptyset$;

* for any $z \in \phi^{-1}(\Delta) \cap \Delta$, $f \circ \phi(z) = f(z)$.

The open convex subset $\mathcal{P}$ and $\phi(\mathcal{P})$ of $\mathbb{C}$ have nonempty intersection. Moreover we saw that no point in $\mathcal{P}$ or $\phi(\mathcal{P})$ can be on a Delaunay segment. The only possibility is that $\phi(\mathcal{P}) = \mathcal{P}$. Then $\phi$ has a fixed point contained in the open polygon $\mathcal{P}$. This contradicts Lemma 5.20. $\square$

The map $\bar{f}$ in not necessarily injective on $\partial\mathcal{P}$, see Figure 40.

**Definition 6.19.** Given a finite type affine surface $(X, \nabla, S)$, we call *exterior* the union of the non-conical Fuchsian singularities and of the image of all affinely immersed half-plane. We call *core* the complement of the exterior.
<figure><img src="image_42.png" /><figcaption>FIGURE 40. The affine surface on the right is formed of two equilateral triangles glued together in a specific way. The disk on the left is immersed in it as indicated. This map injective on the interior of $\mathcal{P}$ but not on its boundary.</figcaption></figure>

The core thus always contains the conical Fuchsian singularities and the irregular singularities and no other singularity. It may be reduced to these singularities, in which case the surface is exceptional, see Section 6.6. Actually we have slightly better: in a non-exceptional surface, every singularity in the core is at the end of at least one Delaunay edge (Lemma 6.22). Reeb cylinders are in the exterior, as are appropriate standard neighborhoods of non-conical singularities and swathes. Note that for translation surfaces, the core is always connected, see [Tah18]. However, for affine surfaces, the core can be disconnected by Reeb cylinders.

**Lemma 6.20.** Consider a disk immersion $(\Delta, f)$ that extends continuously to the boundary. Let $\mathcal{A}$ be a finite set of geodesic segments of $X$, of finite life span (so ending on regular points, conical Fuchsian singularities or foci of irregular singularites). Let $\mathcal{A}$ be the union of the support of the elements of $\mathcal{A}$. Then $f^{-1}(\mathcal{A})$ consists in finitely many segments.

*Proof.* By Lemma 5.9 there are only finitely many points in $\partial\Delta$ mapped to singularities, where there are slit extensions of $f$, and at all the other points there are affine extensions. It follows we can extend $f$ to a neighborhood of $\bar{\Delta}$, minus a finite number of slits form the points mapped to singularities. For the extension, the number of segments is locally finite as it is for any affine immersions. This is also the case near the tip of a slit on $\partial\Delta$, because of the local model of the singularity (which is either conical or a focus of an irregular singularity): finitely many segment ends go straight to it, and they have at most one preimage in the slit in the focus case, and finitely many in the conical case; the other segments stay at some distance of the singularity. $\square$

**Proposition 6.21.** The core is the union of the open Delaunay polygon interiors, the Delaunay segment interiors, the conical Fuchsian singularities and the irregular singularities.

*Proof.* Given a finite type affine surface $(X, \nabla, S)$, assume $x \in X^*$ belongs to the core. Consider a local chart $f_0 : \Delta_0 \to X^*$ with $\Delta_0 = B(z_0, r_0)$ and $f_0(z_0) = x$. Lemma 6.5 ensures that there exists a maximal immersion $(\Delta_1, f_1)$ of a disk/half-plane/plane containing $\Delta_0$ and extending $f_0$. Then $\Delta_1$ must be a disk for otherwise $x$ would belong to the exterior. By Section 6.2, $(\Delta_1, f_1)$ is of type A (disk, flexible) or I (disk, rigid). If it is of type A, this disk defines a particular Delaunay segment $[A, B] \subset \bar{\Delta}_1$. If $z_0 \in [A, B]$, then we are done. Otherwise it is on one side of the segment. By Lemma 6.14 there are only finitely many Delaunay segments. The preimage of these segments in $\Delta_1$ by $f_1$ forms a set $C$ of open chords, including $(A, B)$. This set is finite by Lemma 6.20. Let us pivot the maximal disk towards $z_0$ (see Section 6.3) until we meet a rigid maximal immersion $(\Delta_2, f_2)$. The domain $\Delta_2$ necessarily contains $z_0$. So it cannot be a half-plane, for $z_0$ is not in the exterior. Being rigid, it necessarily defines a Delaunay polygon $P_2$, moreover, $[A, B]$ is one of its edges. If $z_0$ belongs to the closed polygon $P_2$, we are done. Otherwise, it belongs to a part of the disk $\Delta_2$ cut by an edge $[A_2, B_2]$ of $P_2$ that is closer to $z_0$ than
[A, B]. Moreover [A₂, B₂] ∩ Δ₁ is necessarily a chord in the set C. We then repeat the process and pivot the maximal disk Δ₂ to get a new rigid maximal immersion (Δ₃, f₃), etc. At each step n, if z₀ is not in the closed polygon Pₙ, then the side [Aₙ, Bₙ] of Pₙ separating z₀ from the interior of Pₙ in Δₙ, satisfies that [Aₙ, Bₙ] ∩ Δ₁ is one of the chords in C, and moreover these chords get closer and closer to z₀ in the sense that the component of Δ₁ minus the chord that contains z₀ is strictly decreasing for the inclusion. This implies by finiteness of C that the process must stop at some point, i.e. that z₀ ∈ Pₙ for some n. □

We did not state above that the core is the union of Delaunay polygons. Indeed it may
happen that the core has connected components that are graphs, or single points: consider
for instance the whole plane from which a square is removed, and the sides of the square
are glued together (we can get a sphere or a torus according which pairing is chosen). Its
core consist in the two Delaunay segments on the boundary of the removed square.

6.6. **Exceptional affine surfaces.** These surfaces have several characterization, one be-
ing that they have no Delaunay segments. We start with a useful lemma.

**Lemma 6.22.** *Assume a finite type affine surface (X, ∇, S) has a conical Fuchsian singularity s ∈ S with no Delaunay segment ending on it, or an irregular singularity with a focus s with no Delaunay segment ending on it. Then X* is either isomorphic to the exponential-affine plane or to a skew cylinder.*

**Proof.** Every maximal element of the Delaunay category is a half-plane immersion. By the local model of $s$, there exists a disk immersion $(\Delta, f)$ which extends continuously at $\partial\Delta$ (in the case of a focus, we use the extended topology defined in Section 3.7) and such that one point $z_0 \in \partial\Delta$ is mapped to $s$. Consider a maximal Delaunay element $(\Delta', f')$ extending $(\Delta, f)$. Then $\Delta'$ is a half-plane and $z_0 \in \partial\Delta'$. There can be no discontinuity point $z_1 \in \partial\Delta'$ (for the usual topology on $X$) for otherwise we would be in the situation where $f' \circ \phi = f'$ for some dilation $\phi$ fixing $z_1$, and every point of the form $\phi^n(z_0)$, $n \in \mathbb{Z}$, would be mapped to $s$ so there would be, by Lemma 6.15 Delaunay segments obtained by taking the image by the extension of $f'$ of the portion of $\partial\Delta'$ between two consecutive points mapped to $S$. Then, apart from $z_0$, there can not be any other point of $\partial\Delta'$ mapped to $S$ for otherwise there would be a Delaunay segment for the same reason. It means we can pivot $\Delta'$ in the two directions, by a rotation around $z_0$ (see Section 6.3). This rotation goes on forever, in both directions, for an hypothetical half-plane immersion at the end would contradict absence of Delaunay segment for the same reasons as above. This implies that $\tilde{C}^*$ immerses into $X^*$. We then conclude with Proposition 5.6 that $X^*$ either admits an affine immersion of the plane, a situation which is classified in Proposition 5.5, or is isomorphic the exponential-affine plane, to a full Reeb cylinder, to a skew cylinder or to an affine torus. The translation and affine tori have no singularity so are excluded. A whole plane has only one singularity, which is anti-conical Fuchsian, a full translation cylinder has only two, which are cylindrical Fuchsian, and a full Reeb cylinder only two, which are of Reeb type, so these cases are excluded too. □

**Proposition 6.23.** If a finite type affine surface $(X, \nabla, S)$ has no Delaunay segments, then it is isomorphic to one of the following models:

(a) whole plane: $X$ is a sphere, $X^* = \mathbb{C}$ and $\nabla$ has a unique Fuchsian singularity of residue equal to 2;

(b) translation cylinder: $X$ is a sphere, $X^* = \mathbb{C}/\mathbb{Z}$ and $\nabla$ has two Fuchsian singularities with residues equal to 1 (the two ends of the cylinder);

(c) translation torus: $X = \mathbb{C}/\Lambda$ where $\Lambda$ is a lattice of $(\mathbb{C}, +)$ and $\nabla$ has no singularity.

(d) infinite angle cone: $X$ is a sphere, $X^*$ coincides with the universal covering $\mathcal{E}$ of the punctured plane $\mathbb{C}^*$ (also called the exponential-affine plane) and the only singularity of $\nabla$ is a double pole, of residue equal to 2;
(e) affine cylinder (skew cylinder or full Reeb cylinder): $X$ is a sphere and $X^*$ is a quotient of $\mathcal{E}$ by a linear transformation;

(f) affine torus: $X$ is a quotient $\mathcal{E}$ by a lattice and $\nabla$ has no singularity.

*Proof.* If $X$ has a singularity, it is covered by Lemma 6.22 and we are in situation (a), (b), (d) or (e). If $X$ has no singularity then consider a maximal Delaunay category element $(\Delta, f)$. The set $\Delta$ cannot be a disk for there would be a Delaunay edge. If $\Delta$ is a half-plane we are in the type B case (Reeb, flexible) of Definition 6.9. In this case we can proceed exactly as in the proof of Lemma 6.22 and deduce that there is an immersion of the exponential-affine plane $\mathcal{E}$ in $X^*$ and we get the same conclusion, except that case (f) is possible. Finally, Proposition 5.5 classifies the case $\Delta$ is the whole plane, which leads to cases (a), (b) or (c). $\square$

Actually the converse holds. It can be directly seen on each models. Alternatively, it follows from Theorem 1.6 (for instance all the examples satisfy criterion (iii): $2g+n \le 2$).

*Proof of Theorem 1.6.* We have to prove that for a finite type affine surface $(X, \nabla, \mathcal{S})$ of genus $g$ with $n$ poles (counted with multiplicity), the following statements are equivalent:

(o) $X$ is in the list of Proposition 6.23;

(i) every maximal geodesic on $X^*$ is infinite in the future or in the past;

(ii) $X$ has no Delaunay segment;

(iii) $2g+n \le 2$;

(iv) the universal covering of $X^*$ is either the flat plane $\mathbb{C}$ or the exponential-affine plane $\mathcal{E}$;

(v) there exists an affine immersion $\mathcal{E} \to X^*$;

(vi) the core is contained in $\mathcal{S}$.

Let us justify (o) $\Rightarrow$ (iv) $\Rightarrow$ (i) $\Rightarrow$ (ii) $\Rightarrow$ (o), (o) $\Leftrightarrow$ (iii), (iv) $\Leftrightarrow$ (v) and (ii) $\Leftrightarrow$ (vi).

(o) $\Rightarrow$ (iv) follows from inspection of the list: (a), (b), (c) are quotients of $\mathbb{C}$ and (d), (e), (f) of $\mathcal{E}$.

(iv) $\Rightarrow$ (i): every geodesic lifts to the universal covering $\mathbb{C}$ or $\mathcal{E}$. In $\mathbb{C}$, every geodesic have infinite lifespan in the past and future. In $\mathcal{E} \cong \tilde{\mathbb{C}}^*$ every geodesic too, except those aimed at 0, which hit 0 in finite time and are infinite lived in the other direction.

(i) $\Rightarrow$ (ii): every Delaunay segment is a goedoesic whose lifespan is finite in both directions.

(ii) $\Rightarrow$ (o): this is Proposition 6.23.

(o) $\Rightarrow$ (iii) follows by inspection of the list: (a) and (d) $(g, n) = (0, 1)$, (b) and (e) $(g, n) = (0, 2)$, (c) and (f) $(g, n) = (1, 0)$.

(iii) $\Rightarrow$ (o): this is Proposition 2.24.

(iv) $\Rightarrow$ (v): if there is a universal covering $\pi: \mathcal{E} \to X^*$ then this is trivial: $\pi$ is an affine immersion from $\mathcal{E} \to X^*$. If there is a universal covering $\pi: \mathbb{C} \to X^*$ then $\pi \circ \exp: \mathbb{C} \to X^*$ is an affine immersion (where $\exp: \mathcal{E} \to \mathbb{C}^* \subset \mathbb{C}$).

(v) $\Rightarrow$ (iv): see Proposition 5.6 and Proposition 5.5.

(ii) $\Leftrightarrow$ (vi): immediate from the definitions and the fact that Delaunay polygons are bounded by Delaunay segments. $\square$

## 6.7. Classification of components of the Delaunay decomposition.
In this section we prove Theorem 1.7.

Consider a non-exceptional finite type affine surface $(X, \nabla, \mathcal{S})$. By (ii) of Theorem 1.6 there is at least one Delaunay segment. By Corollaries 6.12 and 6.13 the Delaunay segment interiors are embedded and disjoint. Consider the union of all Delaunay segments. By Lemma 6.22, the endpoints contains all conical Fuchsian singularities and all irregular singularities. The converse inclusion holds by definition of Delaunay segments. By
Lemma 6.14 there are finitely many Delaunay segments. Their union is thus a closed subset of X and the complement of this union is an open subset of X*. Its connected components are called the *Delaunay components*. Theorem 1.7 states that these components are of the following types:

(i) **open Delaunay polygon**: defined in Definition 6.16, they are in particular embedded convex polygons with $p \ge 3$ sides drawn between points of $S$;

(iia) **finite angle Reeb cylinder**: dilation cylinders of angle at least $\pi$;

(iib) **Semi-infinite Reeb cylinder** and its end, a Reeb-type Fuchsian singularity;

(iii) **Semi-infinite translation cylinder** and its end, a cylindrical Fuchsian singularity;

(iv) **anti-conical domain** associated to an anti-conical Fuchsian singularity;

(v) **swath domain** associated to an irregular singularity swath.

We refer to Section 2.3 for the description of the cylinders, to Section 4.4 for anti-conical and swath domains and to Section 6.5 for Delaunay polygons.

**Lemma 6.24.** An anti-conical or swath domain is a Delaunay component.

*Proof.* They are bounded by simple saddle connections $\gamma$ that do not cross each other. It is enough to prove that these connections are Delaunay segments. For each $\gamma$, Lemma 4.15 allows to define a half-plane immersion, that will be rigid of type II, and whose boundary extension has a segment mapping to $\gamma$. By Lemma 6.15, $\gamma$ is a Delaunay segment. $\square$

Similarly, any maximal embedded and non-full Reeb cylinder and its eventual end singularity form a Delaunay component, and this is the case too for a maximal embedded non-full translation cylinder and its eventual end singularity.

Let us prove Theorem 1.7. Consider a Delaunay component $C$. Let $x \in C$. If $x$ is in the core of $X$ (defined in Definition 6.19), then Proposition 6.21 proves that $x$ is in an open Delaunay polygon. Otherwise, $x$ is in the exterior of $X$ (the complement of the core). Let $(\Delta, f)$ be a half-plane immersion such that $x \in f(\Delta)$. Without loss of generality, we can assume that $f$ is maximal in the sense of Definition 6.2. These maximal immersions are classified in Section 6.2, see Definitions 6.9 and 6.10. Note that $f(\Delta)$ is contained in the exterior, hence in $C$.

If $f$ is a rigid immersion of type III (translation cylinder), then its image is a semi-infinite translation cylinder open neighborhood of a cylindrical Fuchsian singularity bounded by a chain of saddle connections that are Delaunay segments by Lemma 6.15 (see point Item iii in Proposition 5.23). It follows that $C$ coincides with the translation cylinder and its end.

If $f$ is a flexible immersion of type C (Reeb) or a rigid immersion of type IV (Reeb, opening angle $> \pi$) or $V$ (Reeb, opening angle $\pi$), then Proposition 5.23 shows that its image is contained in a dilation cylinder. Consider the maximal dilation cylinder $C'$ containing $f(\Delta)$. It has two ends, which may be finite or infinite (in terms of the angle span). An infinite end is a neighborhood of a Reeb type Fuchsian singularity by Lemma 2.16. A finite end is necessarily bounded by Delaunay segments by Proposition 5.23 and lemma 6.15. Again, $C$ coincides with the Reeb cylinder.

In the remaining cases, for any $x \in C$, $f$ is either a flexible immersion of type B (non-cylinder) or a rigid immersion of type II (non-cylinder). In these cases, the map $f$ extends continuously to $\partial\Delta$ and $f^{-1}(S)$ consists in finitely many points. They cut $\partial\Delta$ into two half-lines and finitely many segments (0 for type B). We can apply the pivot process described in Section 6.3 with egress component either of the two half-lines. Initially we get type B immersions. In the process we may meet rigid immersions, but note that they have to be of type III because in the other cases, we already classified the situation and they exhibit no type B immersions. We get a continuous family of immersions $(\Delta_\theta, f_\theta)$ parametrized by the pivot angle $\theta \in I$, with $I$ an open interval contained in $\mathbb{R}$ (we will soon prove that $I$
can be taken equal to $\mathbb{R}$). These immersions all map to the exterior, so, by connectedness, to $C$.

Applying Proposition 5.23 to each $(\Delta_{\theta}, f_{\theta})$, we get that we are in situation (i) or (iv) of that statement. In particular, at infinity $f_{\theta}$ tends to a singularity $y \in X$, which is either anti-conical Fuchsian or irregular. Since the $f_{\theta}$ overlap, we gradually get that all $f_{\theta}$ tend to the same $y$ at infinity.

Rigid immersions occur for isolated parameters $\theta$, and each defines one or more, but finitely many, consecutive Delaunay segments sides on $\partial C$. Moreover, two consecutive rigid parameters define consecutive Delaunay segments sides on $\partial C$. Since there are only finitely many Delaunay segment, $C$ is bounded by finitely many Delaunay segment sides (one segment can bound $C$ on at most two of its sides).

**Remark 6.25.** Though not strictly necessary, it is interesting to interpret this in terms of the the affine Delaunay spine introduced in Section 6.3, a graph whose points represent all the maximal immersion equivalence classes. The real number $\theta$ parametrizes a point moving on a succession of edges on the part of the Delaunay spine corresponding to half-plane immersions. We recall that interior points of the edge are the flexible immersions, while vertices are the rigid ones.

There is at least one rigid parameter, for or otherwise by the proof of Lemma 6.22, $X$ would be a skew cylinder, hence exceptional.

Two situations may occur:

- Case 1: there are two different parameters $\theta, \theta'$ such that the associated immersions follow the same side of a given Delaunay segment. Then these immersions are equivalent $(\phi(\Delta_{\theta}) = \Delta_{\theta'}$ and $f_{\theta'} \circ \phi = f_{\theta}$ or some $\phi \in \mathrm{Aut}(\mathbb{C})$).

- Case 2: if we are not in Case 1, this implies that the number of rigid parameters $\theta$ is at most the number of sides of $C$. This implies we can take $I = \mathbb{R}$: indeed after the last rigid immersion, the pivoting only happens around a single point, so at a finite upper/lower bound of $I$, there is an immersion, which can then be still pivoted since it cannot be rigid, leading to a contradiction.

If the function $\theta \in I \mapsto (\Delta_{\theta}, f_{\theta})$ is not injective then using affine continuation we can periodize: we can take $I = \mathbb{R}$ and up to equivalence, the function $\theta \in \mathbb{R} \mapsto (\Delta_{\theta}, f_{\theta})$ is necessarily $\theta_p$-periodic. Hence in both Cases 1 and 2, we can take $I = \mathbb{R}$. And Case 1 is equivalent to the fact that $\theta \in \mathbb{R} \mapsto (\Delta_{\theta}, f_{\theta})$ is periodic up to equivalence.

Let us assume we are in Case 1. Take $\theta_p$ to be the minimal period. Call

$$ \mathcal{U} = \bigcup_{\theta \in [0, \theta_p)} f_{\theta}(\Delta_{\theta}) \subset \mathcal{C} \setminus \{y\} $$

note that if $y$ is an irregular singularity then $y \notin C$; however we will prove in a moment that $y$ is anti-conical, that $C = \mathcal{U} \cup \{y\}$ and that $C$ is an anti-conical domain, as per Definition 4.13. We will also prove as a bonus that $y$ has opening angle $\theta_p$.

There are only finitely many $\theta \in [0, \theta_p)$ such that $(\Delta_{\theta}, f_{\theta})$ is rigid, i.e. such that the extension of $f$ to $\partial\Delta_{\theta}$ passes more that once through a singularity, and there is at least one. Without loss of generality we assume that $\theta = 0$ is one of them. Consider the circular list of rigid parameters in $[0, \theta_p]$. If necessary, insert finitely many (flexible) $\theta$ in the list so that every consecutive elements of the list differ by less than $\pi$. Call $\theta_0 = 0 < \theta_1 < \dots < \theta_{k-1} < \theta_p$ the obtained finite list. For each $\theta_i \neq \theta_p$, orient the line $\partial\Delta_{\theta_i}$ so that $\Delta_{\theta_i}$ is on the left of the line, and denote $A_i$ the first point on $\partial\Delta_{\theta_i}$ that is mapped to a singularity and $B_i$ the last. Then $\mathcal{U}$ decomposes as the union of $f(S_i)$ where $S_i \subset \Delta_i$ is a sector based on $A_i$, with one boundary line going through $B_i$ and the other being the rotation of this line by $\theta_{i+1} - \theta_i \in (0, \pi]$, and of $f(L_i)$ where $L_i \subset \partial\Delta_i$ is the half-line from $B_i$ and onward. See Figure 41. The disjoint union of the $S_i$ and $L_i$ defines an affine
<figure><img src="image_43.png" /><figcaption>FIGURE 41. Finite angle unbounded domain reconstruction.</figcaption></figure>

surface $\mathcal{A}$ that surjectively immerses in $\mathcal{U}$ via the $f_\theta$:

$$F: \mathcal{A} \to \mathcal{U}.$$

Let us show that $F$ is proper from $\mathcal{A}$ to $\mathcal{C} \setminus \{y\}$. Consider a sequence $a_n \in \mathcal{A}$ that leaves every compact subset of $\mathcal{A}$. We must prove that $F(a_n)$ tends to $\partial\mathcal{C} \cup \{y\}$, i.e. that a subsequence tends to a point of that set. If $a_n$ tends to infinity in the $S_i$ or the seams between them then we already know that $F(a_n)$ tends to $y$. Otherwise, a subsequence of $a_n$ tends to a point in one of the segments $[A_i, B_i]$ and by continuous extension of the $f_{\theta_i}$ (and of possibly the adjacent $f_{\theta_j}$) we get the claim.

As we saw, $F$ extends continuously by mapping $\infty$ to $y$, hence holomorphically. In particular, $y$ is separated from $\partial\mathcal{C}$ by an annulus, so $y$ cannot be one of the irregular singularities (which all belong to the union of Delaunay segments), hence $y$ is anti-conical.

As a holomorphic proper map without critical points over a connected set is a finite degree covering, so $F$ is a covering from $\mathcal{A}$ to $\mathcal{C} \setminus \{y\}$. It is hence surjective and $\mathcal{C} \setminus \{y\}$ must be homeomorphic to a covering quotient of the pointed disk, so is to a pointed disk too, so $\mathcal{C}$ is simply connected. It is bounded by a finite chain of Delaunay segments (hence saddle connections), with possible repetitions, and the angle between them is $\ge \pi$ according to the model $\mathcal{A}$.

*Remark 6.26.* We do not stricto sensu need it but we find it interesting to prove that $F$ has degree $\le 1$, so is injective, hence an isomorphism from $\mathcal{A}$ to $\mathcal{C} \setminus \{y\}$. By the same argument as above, if $x \in \mathcal{C} \setminus \{y\}$ is close to $y$ then all the elements in $F^{-1}(\{x\})$ are close to $\infty$. A neighborhood of $\infty$ in $\mathcal{A}$ is isomorphic to an anti-conical standard neighborhood with angle $\theta_p$ and ratio some $s > 0$, with possibly a shift (see shifted anti-conical singularities in Section 2.4). If $\deg F > 1$ then the deck transformation generator of the covering $F$ necessarily is a rotation of angle $\theta_p/ \deg F$ (a priori it may be $> 2\pi$ and it may include a translation in shifted case). But then this would imply that the period of the family $(\Delta_\theta, f_\theta)$ up to equivalence, is strictly smaller than $\theta_p$, in contradiction with the minimality of $\theta_p$.
<figure><img src="image_44.png" /><figcaption>FIGURE 42. Proof of Lemma 6.27</figcaption></figure>

Let us now assume we are in Case 2. No two $(\Delta_{\theta}, f_{\theta})$, $\theta \in \mathbb{R}$, are equivalent. We can still define an affine surface $\mathcal{A}$ with an immersion $F : \mathcal{A} \to \mathcal{C}$. This times it needs infinitely many flexible $\theta_i$, more precisely: finitely many between rigid parameters; after the last and before the first, an infinite sequence that we can take spaced by $\pi/2$ for instance. Let $\theta_i$, $i \in \mathbb{Z}$ the increasing bi-infinite sequence thus obtained.

**Lemma 6.27.** Let $i < j$ be such that $\theta_j < \theta_i + \pi$. Then $S_i \cup L_{i+1}$ is disjoint from $S_j \cup L_{j+1}$.

*Proof.* If $j = i + 1$ this is easy. Otherwise, we proceed as follows: the segment $[B_k, A_k]$ is supported on $\partial\Delta_{\theta_k}$. We have $B_k = A_{k-1}$ for all $k \in \mathbb{Z}$. The vector $A_k - B_k = A_k - A_{k-1}$ has argument $\theta_k$. The vectors $A_k - B_k$ for $k$ between $i$ and $j-1$ included, have an argument in $(\theta_j - \pi, \theta_j]$. This implies that the distance from $A_k$ to $\Delta_{\theta_j}$ is strictly increasing as $k$ decreases from $j-1$ to $i-1$. In particular, the closure of the set $S_i$, which is a closed sector based on $A_i$ and whose elements are linear combination with non-negative coefficients of the vectors $A_{i-1} - A_i$ and $A_i - A_{i+1}$ lies in the complement of $\bar{\Delta}_{\theta_j}$. $\square$

Let us prove that $F$ is injective. By contradiction assume there is some $i, i' \in \mathbb{Z}$, $z \in S_i \cup L_{i+1}$ and $z' \in S_{i'} \cup L_{i'+1}$ such that $f_{\theta_i}(z) = f_{\theta_{i'}}(z')$ but $z \neq z'$ or $i \neq i'$. There exists $\phi \in \operatorname{Aut}(\mathbb{C})$ such that $\phi(z') = z$ and such that $\forall w$ close to $z$, $f_{\theta_i} \circ \phi(w) = f_{\theta_{i'}}(w)$. The map $f_{\theta_i} \circ \phi$ is defined on the half-plane $H = \phi^{-1}(\Delta_{\theta_i})$ and is a rigid maximal Delaunay element, since its continuous extension maps at least two boundary point to a singularity. There is a unique $\theta \in [\theta_{i'} - \pi, \theta_{i'} + \pi)$ such that $H$ is oriented like $\Delta_{\theta}$, i.e. is a translate thereof. We cannot have $\theta = \theta_{i'} - \pi$ for otherwise, it would allow to extend $f_{\theta_{i'}}$ to the whole plane, and $(X, \nabla, S)$ would be exceptional. The two sets $H$ and $\Delta_{\theta}$ they are either equal or one contains strictly the other. By affine continuation, the map $f_{\theta_i} \circ \phi$ coincides $f_{\theta_{i'}}$ with on the infinite sector $H \cap \Delta_{\theta_{i'}}$. By definition, the map $f_{\theta}$ coincides $f_{\theta_{i'}}$ on the infinite sector $\Delta_{\theta} \cap \Delta_{\theta_{i'}}$. It follows that $f_{\theta_i} \circ \phi$ and $f_{\theta}$ coincide on a non-empty open set, hence on the smallest of $H$ and $\Delta_{\theta}$. By maximality of the immersions $f_{\theta_i} \circ \phi$ and $f_{\theta}$, this implies that $H = \Delta_{\theta}$ and $f_{\theta_i} \circ \phi = f_{\theta}$, i.e. $\phi$ is an equivalence between $(\Delta_{\theta}, f_{\theta})$ and $(\Delta_{\theta_i}, f_{\theta_i})$. By hypothesis, this means $\theta = \theta_i$. So $\phi(\Delta_{\theta_i}) = \Delta_{\theta_i}$ and $f_{\theta_i} \circ \phi = f_{\theta_i}$. This implies that $\phi$ fixes $A_i$ and $B_i$ hence $\phi$ is the identity, so $z = z'$, hence $i \neq i'$ by hypothesis. Moreover by one of the above claims, $|\theta_i - \theta_{i'}| < \pi$. So the sets $S_i \cup L_{i+1}$ and $S_{i'} \cup L_{i'+1}$ are disjoint by Lemma 6.27, which contradicts $z = z'$. This proves the claim.

We then prove that $y$ is an irregular singularity. Otherwise, it would be anti-conical Fuchsian. If $y$ is pure anti-conical, consider a pointed neighborhood $W$ of $y$ isomorphic to
a standard neighborhood $V$ of a skew cylinder $\tilde{C}^*/\langle\lambda\rangle$ via an isomorphism $\psi: V \to W$. The immersion $f_0$ maps $z \in \Delta_0$ to $W$ as soon as $|z|$ is big enough. Fix a real $M > 0$. It follows by affine continuation that we can factor $F$ as $F = \psi \circ \pi \circ \Phi$ on a subset $U \subset A$ with: $U$ is the union of $(S_i \cup L_i) \setminus \bar{B}(0, R)$ for the indices $i$ satisfying $\theta_i \in [-M, M]$, $R > 0$, $\Phi: U \to \tilde{C}^*$ is affine, $\pi: \tilde{C}^* \to \tilde{C}^*/\langle\lambda\rangle$ is the canonical projection. If $M$ is chosen big enough, this contradicts injectivity of $F$. In the shifted anti-conical case, the proof is similar.

We then claim that $F$ is proper from $A$ to $C$. As in Case 1, we already know that $\hat{z} \in A$ tends to infinity within one $S_i \cup L_i$ then $F(\hat{z})$ tends to $y$, and that if $\hat{z}$ tends to a boundary segment of $A$ for one of the finitely rigid $\theta_i$, then $F(\hat{z})$ tends to a Delaunay segment. Compared to Case 1 there is a supplementary possibility up to extraction for a sequence of points $\hat{z} \in A$ leaving every compact subset: that $\theta_i$ tends to $\pm\infty$ where $i$ is the index such that $\hat{z} \in S_i \cup L_i$. To treat that case we use that a neighborhood of $\infty$ in $\Delta_0$ maps to a neighborhood of type $B_{n+1/2}$ of the swath, in the nomenclature of Section 3.3.3. This allows by analytic continuation to prove that for some $M > 0$ big enough, for all $i$ such that $\theta_i > M$, $F(S_i \cup L_i)$ maps to the $A_n$ piece and that as $\theta_i \to +\infty$, $F(\hat{z})$ tends to $y$. A similar statement holds for $\theta_i$ tending to $-\infty$. We also get that the first and last points on $\partial A$ map to adjacent foci of $y$ (the same focus if $y$ is a degree 2 pole of $\nabla$), and that $F(A)$ contains a neighborhood of a swath of $y$, and no neighborhood of any other swath of any irregular singularity. Moreover we get that the closure of $F(A)$ consists in $F(A)$ union the image of all boundary $[A_i, B_i]$ of $A$ by the extensions of the finitely many rigid $f_{\theta_i}$.

A proper injective holomorphic map over a connected set is a homeomorphism, so $C$ is isomorphic to $A$, so it is simply connected. From the model of $A$, it follows that the angles at the singularities are $\ge \pi$. We have proved that $C$ a swath domain, as per Definition 4.13.

In the classification we just proved, we have in every case a precise enough description so that we know an exterior Delaunay component $D$ that is not a finite angle Reeb cylinder contains the neighborhood of exactly one swath or non-conical singularity and is disjoint from small enough neighborhoods of all the others, while a finite angle Reeb cylinder is disjoint from small enough neighborhoods of all swathes and non-conical singularities. (If the boundary of $D$ contains an irregular singularity $p$, disjointness of $D$ from small enough swath neighborhoods of $p$ (except for the swath associated to $D$ if $D$ is a swath domain) follows from the local model given in Section 3.3.3 and the fact that the angle of the domain boundary at $p$ is finite).

Conversely, near any anti-conical, Reeb or cylindrical Fuchsian singularity, or irregular singularity swath, we can immerse a half-plane. The Delaunay component $D$ that contains the image of this immersion is necessarily an exterior domain. Every neighborhood of the singularity or swath we started from intersects $D$. By the previous paragraph, $D$ is not a finite angle Reeb cylinder, and thus is associated to a singularity or swath which, again by the previous paragraph, has to be the one we started from.

### 6.8. A complexity bound.
The number of components in the Delaunay decomposition of a finite type affine surface $(X, \nabla, S)$ is controlled by topological data: the number $n$ of poles of $S$ (counted with multiplicity) and the genus $g$ of the underlying Riemann surface $X$. We assume that $2g + n \ge 3$, so that there are Delaunay segments. Theorem 1.10 is actually finer than just a bound on the number of components. It gives an equality, which to be expressed requires to decompose Delaunay polygons without adding vertices, which is always possible. A Delaunay polygon with $s$ sides will give $s - 2$ triangles. Some of the triangles may be self-folded but we allow this. We call $t$ the total number of triangles. Concerning exterior domains, we count the total number of sides on all these domains and
call it $\beta$. Then Theorem 1.10 states that

$$t + \beta = 4g - 4 + 2n.$$

*Proof of Theorem 1.10.* For a non-exceptional finite type affine surface $(X, \nabla, \mathcal{S})$, we draw the union $\mathcal{A}$ of the Delaunay segments. The set $X \setminus \mathcal{A}$ decomposes according to Theorem 1.7. We draw additional arcs to decompose further Delaunay polygons into topological triangles $t$. We obtain a system of arcs $\mathcal{A}'$ (containing $\mathcal{A}$) satisfying the hypotheses of Lemma 4.4. The special faces of the decomposition are exactly the semi-infinite Reeb cylinders, semi-infinite translation cylinders, anti-conical domains and swath domains, together with their singularity (except in the swath domain). The ordinary faces of the decomposition form a topological triangulation of the union of the Delaunay polygons and Reeb cylinders of finite angle. From Lemma 4.4 we obtain:

$$|\mathcal{A}| = 2g - 2 + n + o \\ \text{and } |\mathcal{A}| = 6g - 6 + 3n - \sigma$$

where $o$ is the number of ordinary faces and $\sigma$ the total side count of the special faces. A finite angle embedded Reeb cylinder with side-count $s$ is triangulated by exactly $s$ ordinary faces, whence $o + \sigma = t + \beta$. Combining these identities, we obtain $t + \beta = 4g - 4 + 2n$. $\square$

## 7. EXISTENCE OF MEROMORPHIC CONNECTIONS WITH PRESCRIBED IRREGULAR SINGULARITIES

### 7.1. A Riemann-Hilbert problem.
Meromorphic abelian differentials,<fnref n="23" /> are equivalent to translation surfaces with polar domains. A complete system of local invariants are given by the orders of the zeroes and the order and residues at the poles. The complete characterization of the profiles of local invariants that can be realized by such a differential on a Riemann surface of genus $g$ was given in [GT21]. Analogous results were proven for quadratic differentials<fnref n="24" /> and higher order differentials<fnref n="25" /> in [GT25] and [GT22], respectively.

All these differentials define affine structures corresponding to special subclasses of Fuchsian meromorphic connections<fnref n="26" /> on the tangent bundle of a Riemann surface. In each case, the list of obstructions is remarkably rich, and the algorithms for constructing flat surfaces with prescribed local invariants are highly intricate.

The global realization problem for a profile of local invariants on affine surfaces is a Riemann-Hilbert-type problem, one that has never been solved at the above of generality, let-alone in the presence of irregular singularities:

**Problem.** Given a profile of local analytic invariants for poles, does there exist a Riemann surface $X$ of genus $g$ and a meromorphic connection $\nabla$ on the tangent bundle $TX$ of $X$ whose poles realize this profile of local invariants?

In this paper, we address a significantly simpler problem.

**Problem.** Given a double pole with prescribed local invariants, what is the minimal number $m$ such that there exists a meromorphic connection $\nabla$ on the tangent bundle of $\mathbb{CP}^1$ such that $\nabla$ has a double pole realizing this profile of local invariants, in Fuchsian singularities and no other singularity?

Let us introduce the following terminology. We recall that the asymptotic value family of an irregular singularity is a $\mathbb{Z}$-indexed sequence $u_n$ of complex numbers defined in Proposition 3.3.

<footnote n="23">²³Which take the form $f(z)dz$ in charts, $f$ meromorphic.</footnote>

<footnote n="24">²⁴Which take the form $f(z)dz^2$ in charts.</footnote>

<footnote n="25">²⁵Which take the form $f(z)dz^q$ in charts for some integer $q \ge 3$.</footnote>

<footnote n="26">²⁶With Christoffel symbol $\Gamma(z) = f'(z)/qf(z)$ in charts.</footnote>
**Definition 7.1.** An irregular singularity is centered when the asymptotic value family of a developing map (hence of every developing map) is constant.

According to Section 3.1.2, for a double pole of and a given residue res ∈ C, there are exactly two possible local affine isomorphism classes, and they can be distinguished by the pole being centered or not. The choice of the invariant $\iota \in \mathcal{I}_{2,\text{res}}$, which characterizes the class for a fixed residue, amounts to the choice between centered or not.

**Remark 7.2.** When res ∈ Z, a double pole is centered if and only if b = 0, where the affine holonomy of a developing map of the pole is $L(z) = z + b$. See Definition 1.2.

**Theorem 7.3.** Given a residue res ∈ C and an asymptotic values invariant $\iota \in \mathcal{I}_{2,\text{res}}$, the minimal number s ∈ N such that there exists a meromorphic connection on the tangent bundle of $\mathbb{CP}^1$ with s simple poles and a double pole of residue res and invariant $\iota$ is:

* 0 if res = 2 and $\iota$ is centered;

* 1 if res ∈ Z ∩ [3, +∞) and $\iota$ is centered;

* 1 if res ∈ C \ {2, 3, ...} and $\iota$ is not centered;

* 2 if res ∈ Z ∩ [2, +∞) and $\iota$ is not centered;

* 2 if res ∈ C \ {2, 3, ...} and $\iota$ is centered;

We can present this list differently:

<table><thead><tr><th>res:</th><th>C \ {2, 3, ...}</th><th>2</th><th>3, 4, ...</th></tr></thead><tbody><tr><td>centered</td><td></td><td></td><td></td></tr><tr><td>yes</td><td>2</td><td>0</td><td>1</td></tr><tr><td>no</td><td>1</td><td>2</td><td>2</td></tr></tbody></table>

In the remainder of Section 7, we prove Theorem 7.3 by providing the obstructions in Section 7.2 and explicitly constructing the connections in Section 7.4. Note that the case s = 0 is already completely characterized as the exponential-affine plane $\mathcal{E}$ by Lemma 2.21.

The analogous problem for poles of order $d \ge 3$ is significantly more complicated due to the wide variety of possible asymptotic sequences.

**7.2. Obstructions.** Here we identify and prove obstructions to the existence of a meromorphic connection on $\mathbb{CP}^1$ with one double pole and a small number of simple poles (Fuchsian singularities). Recall (see Proposition 2.10) that the sum of residues of a meromorphic connection on $\mathbb{CP}^1$ is equal to 2.

**Proposition 7.4.** We assume that there exists a meromorphic connection $\nabla$ on $\mathbb{CP}^1$ with exactly two poles, one being a simple pole while the other is a double pole of residue res ∈ C \ {2} and invariant $\iota \in \mathcal{I}_{\text{res},2}$. Then the invariant $\iota$ is centered if and only if res ∈ Z ∩ [3, +∞).

**Geometric proof of Proposition 7.4.** We assume that such a connection exists. Call p the double pole and q the simple pole. The surface is non-exceptional since $(g, n) = (0, 3)$ in the notations of Theorem 1.6. Call

$$ \rho = 2 - \text{res} $$

the residue of the simple pole p. The complement U of the two poles in $\mathbb{CP}^1$ has fundamental group $\pi_1(U)$ isomorphic to Z. Let $\gamma$ be a loop generating it. Let $\pi: \tilde{U} \to U$ be a universal covering. Its deck transformation group is isomorphic to $\pi_1(U)$, hence to Z. Let $\tau$ be the generator associated to $\gamma$. Consider a developing map $\phi: \tilde{U} \to C$. Then according to Section 2.2, $\phi \circ \tau = L \circ \phi$ for some $L \in \text{Aut}(C)$ (called the holonomy) with $L(z) = az + b$ and $a = \exp(-2\pi i \text{res}) = \exp(2\pi i \rho)$ or $a = \exp(2\pi i \text{res}) = \exp(-2\pi i \rho)$ depending on the choice of generator $\gamma$.

If the simple pole q is conical, then it has a punctured neighborhood V affine isomorphic to the following local model: a connected neighborhood of 0 in a closed sector in C (or in
$\tilde{C}^*$ if the angle of the sector needs to be > $2\pi$), based on 0, with the two boundary lines identified by an affine map fixing 0. The developing map sends any lift of $V$ in $\tilde{U}$ to an affine image of the sector. Choose one lift and call $z_0$ the tip of the sector. Then $L$ sends one of the half-lines bounding the sector to the other one, in particular it fixes $z_0$.

If $\mathrm{res} \in \mathbb{Z} \cap [3, +\infty)$, then $\rho \in \mathbb{Z} \cap (-\infty, -1]$. The simple pole is therefore a conical singularity of angle $(1 - \rho)2\pi$. The coefficient $a$ of the affine holonomy $L$ is equal to 1. Moreover, we get from the local model of a conical singularity that the affine holonomy of a small loop around it always has a fixed point. So $L$ is the identity. But $L$ is also the affine holonomy of a small loop winding once around the double pole $q$ (in the other direction). So $q$ is centered by Remark 7.2.

Below we use the Delaunay decomposition of $RS$ associated to the affine structure. The
possible cases that we will meet are illustrated on Figure 43.

If $\mathrm{Re}(\mathrm{res}) > 1$ but $\mathrm{res} \notin \mathbb{Z}$, then we have $\mathrm{Re}(\rho) < 1$ so the simple pole is a conical singularity. Since $\mathrm{Re}(\rho) \notin \mathbb{Z}$, the coefficient $a$ of the affine holonomy $L$ is not equal to 1.

In the Delaunay decomposition induced by the connection (see Theorem 1.7), there
is exactly one swath domain $D$ (containing the swath of the double pole). Following
Theorem 1.10, we have $t + \beta = 2$ where $t$ is the number of triangles in any triangulation
of the rest of the surface with vertices in $S$ while $\beta$ is the number of boundary sides of the
exterior domains (including $D$). There are two cases.

If $\beta = 1$, then $D$ is the only exterior domain and the double pole is non-centered because, following a developing map along the unique Delaunay segment bounding $D$ gives a segment, whose two endpoints are different, which shows that the double pole $q$ has at least two different asymptotic values (the path defined by the Delaunay segment can be shrunk within the closure of $D$ in $X^*$ to an arbitrarily small neighborhood of the double pole, while still preserving an initial and final portion of the path). So $q$ is not centered. We are in the bottom right situation of Figure 43.

If $\beta = 2$, then $D$ is still the only exterior domain because the only Fuchsian singularity is conical and a Reeb cylinder of finite angle would have at least two boundary sides. It follows that $t=0$ and the swath domain has two boundary sides and we obtain the affine surface by taking a model $\mathcal{A}$ as in Case 2 of the proof of Theorem 1.7 in Section 6.7, gluing on each other the two boundary segments of $\mathcal{A}$, corresponding to the two sides of the Delaunay segment, with one end being the conical singularity, while the other end is the focus of the irregular singularity. Following the developing map $\phi$ from the simple to the double pole along this segment, we trace a straight segment in $\mathbb{C}$ from the fixed point $z_0$ of $L$ to a term $u_n$ of the asymptotic value family of the double pole $q$. Hence $L(u_n) \neq u_n$. But $L(u_n) = u_{n+1}$ (or $u_{n-1}$ according to the choices of generators). It follows that $q$ is non-centered. We are in the middle right situation of Figure 43.

If $\mathrm{Re}(\mathrm{res}) \le 1$, then $\mathrm{Re}(\rho) \ge 1$, so the Fuchsian singularity $p$ is cylindrical, of Reeb-type or anti-conical, so no Delaunay segment can reach $p$, and $p$ belong contained to a component $D_1$ of the Delaunay decomposition that is either a semi-infinite translation cylinder, a Reeb cylinder an anti-conical domain or a swath domain (together with their singularities in the non-swath case). Following Theorem 1.7, the Delaunay decomposition of the affine structure defined by the connection must also contain a swath domain $D_2$ containing the swath of the double pole and different from $D_1$. So the total number $\beta$ of boundary sides of these domains satisfies $\beta \ge 2$. Theorem 1.10 tells that $t+\beta=2$ where $t$ is some non-negative integer related to the other domains $D_i$ of the decomposition, $i \ge 3$, if there are. It follows that $\beta=2$ and $t=0$, so there are in fact no domains beyond $D_1$ and $D_2$. The surface is formed by these two domains glued along the two sides of a single closed saddle connection from the double pole to itself. We are in the top right situation of Figure 43. Then the value of the developing map $\phi$ along any lift of the Delaunay segment
to $\tilde{U}$ (the universal covering of the complement in $\mathbb{CP}^1$ of the two poles) follow a straight segment whose ends are two consecutive terms of the asymptotic value family associated to the double pole $q$. Consequently, $q$ is not centered.

Coming back to the case $\mathrm{res} \in \mathbb{Z} \cap [3, +\infty)$, which we saw is centered, the same argument as for the case $\mathrm{Re}(\mathrm{res}) > 1$ implies that the domain $D$ is 2-sided so the Delaunay graph is a single simple curve between the two poles: we are in the middle right situation of Figure 43. $\square$

<figure><img src="image_45.png" /><figcaption>FIGURE 43. The three kind of Delaunay decompositions we can meet for a meromorphic connection as in Proposition 7.4. The Delaunay components (connected components of the complement of the graph) are labeled A, B, C, and this is related to their classification (i), (ii), . . . (v) as in Theorem 1.7 as follows: A for infinite angle unbounded type (v); B for either semi-infinite Reeb (ii) or translation (iii) cylinder, or finite angle unbounded type (iv); C for open Delaunay polygon (i).</figcaption></figure>

**7.3. Sector grafting.** We introduce a surgery that allows us to modify the residues of the singularities.

Assume we are given, on a finite type affine surface $(X, \nabla, S)$, a geodesic $\gamma : (0, +\infty) \to X^*$ such that:

• $\gamma(t)$ converges to a conical singularity $S_1$ of residue $\mathrm{res}_1$ as $t \to 0$;

• $\gamma(t)$ converges to a singularity $S_2$ of residue $\mathrm{res}_2$ as $t \to +\infty$;

• $\gamma$ is injective (not self-crossing).

We make no assumption on the order of the pole at $S_2$ but note that $S_2$ cannot be a conical Fuchsian singularity, for it only would only take finite time for a geodesic to reach it, nor a Reeb type singularity, which is unreachable.

We cut $X$ along the line traced by $\gamma$: this gives a surface with boundary, the slit having a left side and a right side with respect to the orientation of $\gamma$. We then glue an infinite sector $C_\theta$ of angle $\theta \in (0, +\infty)$, $C_\theta = \{z \in \tilde{\mathbb{C}}^*; \arg z \in [0, \theta]\}$, as follows: a point in $C_\theta$ of polar coordinate $(r, 0)$ is glued to the right boundary point of the slit at $\gamma(r)$; the point of $C_\theta$ of polar coordinate $(sr, \theta)$ is glued to the left boundary point of the slit at $\gamma(r)$, where $s \in \mathbb{R}_{>0}$ is some dilation factor. See Figure 44.

The obtained topological space is again a surface, and actually a finite type affine surface $X'$. The conical point remains a conical point and $S_2$ remains a singularity, of the same polar order. In $X'$, the residues at the singularities $S_1$ and $S_2$ become respectively

$$ \mathrm{res}'_1 = \mathrm{res}_1 - \frac{\log(s) + i\theta}{2\pi i} = \mathrm{res}_1 + i\frac{\log(s)}{2\pi} - \frac{\theta}{2\pi} \quad \text{and} $$

$$ \mathrm{res}'_2 = \mathrm{res}_2 + \frac{\log(s) + i\theta}{2\pi i} = \mathrm{res}_2 - i\frac{\log(s)}{2\pi} + \frac{\theta}{2\pi}. $$
<figure><img src="image_46.png" /><figcaption>FIGURE 44. Grafting on a torus (Section 7.3)</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>FIGURE 45. Grafting an infinite angle seen as a limit process.</figcaption></figure>
This operation does not affect the other singularities.

We can also graft an infinite angle, by grafting two semi exponential half-planes to each side of the slit $\gamma$. In other words, cut $\tilde{C}^*$ into two pieces along the radial line of argument 0. Glue the top piece to the left of the slit and the bottom piece to the right of the slit. This time there is no dilation parameter, and the two singularities merge into a single singularity $S'$, whose order is $d+1$ where $d$ is the order of $S_2$.

**Lemma 7.5.** *The residue of $S'$ is $\text{res}_1 + \text{res}_2$.*

*Proof.* Take a simple closed loop near $\gamma$ winding anticlockwise around it: its holonomy is $\exp(-2\pi i(\text{res}_1 + \text{res}_2))$ and it does not change after the grafting, then using that the linear holonomy only depends on free homotopy class, we get $\exp(-2\pi \text{res}') = \exp(-2\pi i(\text{res}_1 + \text{res}_2))$ so $\text{res}' \equiv \text{res}_1 + \text{res}_2 \bmod \mathbb{Z}$. To get rid of the modulo, one looks at the turning number of the loop, as defined in Section 2.2 and the deformation invariance stated in Lemma 2.7 to reduce to a (non-injective) $C^1$ curve that loops around $S_1$, then rides along $\gamma$ then loops around $S_2$, then rides back; the small loops have classical turning number 0 in a chart, not 1, so eq. (2.9) gives that their turning number for the affine structure is $-\text{Res}(\text{not } 1-\text{Res}$ as in eq. (2.10)).

Alternatively one could try and prove convergence of the finite angle grafting (with no dilation) to the infinite angle grafting and use the computation of residues we made in the finite angle case. See Figure 45. $\square$

Informally, the conical singularity becomes a focus, while for the other singularity: if it is Fuchsian, it becomes a swath; if it is irregular, the swath to which $\gamma$ converges before the infinite angle grafting, is cut into two swathes.

## 7.4. Construction of connections with prescribed irregular singularity class.

### 7.4.1. *Constructions with one additional Fuchsian singularity.*
The first construction consists in gluing together, along a closed segment that will end up being a saddle connection, two domains:

* a swath domain<fnref n="27" /> that will contain a swath of the double pole. It is constructed by taking the universal covering $\tilde{C}^*$ of $C^*$, slitting it along a half-line from 0, and inserting a semi-infinite strip. See the bottom of Figure 46.

* An anti-conical domain or a semi-infinite translation cylinder or a semi-infinite Reeb cylinder (an exterior domain containing the simple pole). See the top part of Figure 46.

Both ends of the segment tend to the focus the double pole. In this construction, the residue of the simple pole can be freely chosen in $\{\rho \in \mathbb{C}; \text{Re}(\rho) \ge 1\}$. Then the residue res at the double pole can be any complex number satisfying $\text{Re}(\text{res}) \le 1$ since $\text{res} + \rho = 2$ by Proposition 2.10. Note that in this construction, the double pole is never centered.

In the second construction, for any $\rho$ with $\text{Re}\rho < 1$, start from a skew cylinder where the conical singularity has residue. Take a half-line embedded in the regular part, for instance a radial line minus an initial portion from the conical singularity. One end of this geodesic is a regular point, the other is the anti-conical point at infinity. Perform an infinite angle sector grafting as in Section 7.3. This merges the regular and anti-conical points into a double pole, without changing the conical singularity, and the resulting compact Riemann surface is still homeomorphic to a sphere.

For each of these two constructions, we can either use Proposition 7.4 to determine whether the double pole is centered or not, depending on the value of res, or see it directly on the geometric model.

<footnote n="27">²⁷Using the terminology of Section 6.7. It is not necessary to have read that section to understand the construction here.</footnote>
<figure><img src="image_48.png" /><figcaption>FIGURE 46. First and second construction of Section 7.4.1</figcaption></figure>

<figure><img src="image_49.png" /><figcaption>FIGURE 47. One way to present the second construction of Section 7.4.1.</figcaption></figure>

<footnote n="7">7.4.2. *Constructions with a non-centered double pole and two additional Fuchsian singularities.* We have to construct a meromorphic connection on $\mathbb{CP}^1$ with a non-centered double pole of residue $res \in \mathbb{Z} \cap [2, +\infty)$. We start from the first construction of Section 7.4.1 where for the second domain we chose a semi-infinite translation cylinder. We then take any point in the first domain draw a half-line to infinity within that domain (see an example on Figure 48), on which we perform a grafting (see Section 7.3) of angle $2\pi(res - 1)$ and no dilation. We obtain a meromorphic connection on $\mathbb{CP}^1$ with a Fuchsian singularity of residue 1 at the end of the translation cylinder and a Fuchsian singularity of residue</footnote>
<figure><img src="image_50.png" /><figcaption>FIGURE 48. The construction of Section 7.4.2 consists in applying the sector grafting of Section 7.3 to the first construction of Section 7.4.1 (with a translation cylinder for the second domain). The sector is inserted along the line indicated by scissors and has an angle that is a positive integer number of turns.</figcaption></figure>

1-res, being a conical singularity of angle $2\pi$ res. Then, the double pole has a saddle self-connection so is non-centered and its residue is equal to res since the sum of the three residues has to be 2.

<figure><img src="image_51.png" /><figcaption>FIGURE 49. The construction of Section 7.4.3 consists in applying two sector graftings to the first construction of Section 7.4.1 (with a translation cylinder for the second domain). The sectors are inserted along: the line indicated by scissors; the vertical line along which a strip was initially glued to give a translation cylinder.</figcaption></figure>
<figure><img src="image_52.png" /><figcaption>FIGURE 50. Delaunay decompositions corresponding to the constructions of Sections 7.4.2 and 7.4.3. Same conventions as on Figure 43.</figcaption></figure>

### 7.4.3. Constructions with a centered double pole and two additional Fuchsian singularities.
We have to construct a meromorphic connection on $\mathbb{CP}^1$ with a centered double pole of residue $\rho \notin \mathbb{Z} \cap [2, +\infty)$. We also start here from the first construction of Section 7.4.1 where for the second domain we chose a semi-infinite translation cylinder. We then take the mid-point of the saddle connection between the two domains and draw a vertical half-line to infinity within the first domain (see on Figure 49), on which we perform a grafting of angle $2\pi k - \pi$ and no dilation, $k \in \mathbb{N}^*$. We also perform a grafting on the vertical line bounding the upper half-strip (initially glued to the other vertical side to give a translation cylinder), inserting an angle of $2\pi a$, $a \in [0, +\infty)$ and this time allowing for a dilation of factor $s > 0$. We obtain a meromorphic connection on $\mathbb{CP}^1$ with:

* at infinity in the former cylinder, a Fuchsian singularity of residue $\rho_1$ with $\rho_1 = 1 + a + i \frac{\log s}{2\pi}$;

* at the midpoint, a Fuchsian singularity of residue $1/2 - k$, since it is conical with no dilation and angle $2\pi k + \pi$.

By construction, the double pole is centered. Its residue is equal to

$$ \mathrm{res} := k + \frac{1}{2} - a - i \frac{\log s}{2\pi} $$

since the three residues add up to 2. Since $k$ and $a$ can be chosen arbitrarily in $\mathbb{N}^*$ and $[0, +\infty)$ respectively, $\mathrm{Re}(\mathrm{res})$ can take any prescribed value. Finally $\mathrm{Im}(\rho)$ can take any prescribed value since $s$ can be freely chosen in $(0, +\infty)$

### 7.5. Analytic approach and Euler's Gamma function.
Here we reprove part of Theorem 7.3 with another approach. More precisely we get the minimal value $s$ on the number of supplementary Fuchsian singularities when this value is 0 or 1 but instead of $s = 2$ we only get $s \ge 2$. Interestingly, part of the argument is based on Euler's Gamma function.

**Definition 7.6.** Let $\mathcal{C}$ be the class of finite type meromorphic connections on $\mathbb{CP}^1$ with two singularities: a double pole and either a simple pole or a marked point.

Up to a homographic change of variable we can assume that the double pole is at 0 and that the other singularity is $\infty$.

We denote here $\zeta$, instead of $\Gamma$, the Christoffel symbol of the connection in the canonical chart $\mathbb{C}$ of $\mathbb{CP}^1$. It is a holomorphic function on $\mathbb{C}^*$. It thus has a Laurent power series expansion

$$ \zeta(z) = \sum_{n \in \mathbb{Z}} a_n z^n $$

with domain of convergence containing $\mathbb{C}^*$. By hypothesis, it has at 0 a double pole, so the sum may be restricted to $n \ge -2$. Under the change of variable $w = 1/z$, the Christoffel symbol transforms into

$$ \zeta_w = -\frac{1}{w^2} \zeta(1/w) - \frac{2}{w} = -\frac{2}{w} - \sum_{n \ge -2} a_n w^{-n-2} $$