# POLYGONAL SURFACES IN PSEUDO-HYPERBOLIC SPACES

Alex Moriani

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Alex Moriani. POLYGONAL SURFACES IN PSEUDO-HYPERBOLIC SPACES. 2024. hal-04863124

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# POLYGONAL SURFACES IN PSEUDO-HYPERBOLIC SPACES

ALEX MORIANI

ABSTRACT. A polygonal surface in the pseudo-hyperbolic space $H^{2,n}$ is a complete maximal surface bounded by a lightlike polygon in the Einstein universe $Ein^{1,n}$ with finitely many vertices. In this article, we give several characterizations of them. Polygonal surfaces are characterized by finiteness of their total curvature and by asymptotic flatness. They have parabolic type and polynomial quartic differential. Our result relies on a comparison between three ideal boundaries associated with a maximal surface, corresponding to three distinct distances naturally defined on the maximal surface.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. General context</td><td>2</td></tr><tr><td>1.2. Main theorem</td><td>2</td></tr><tr><td>1.3. Related works</td><td>3</td></tr><tr><td>1.4. Overview of the proof</td><td>5</td></tr><tr><td>1.5. Organization of the paper</td><td>7</td></tr><tr><td>Acknowledgments</td><td>8</td></tr><tr><td>2. Preliminaries</td><td>8</td></tr><tr><td>2.1. The pseudo-hyperbolic space</td><td>8</td></tr><tr><td>2.2. Spacelike surfaces</td><td>10</td></tr><tr><td>2.3. The ideal boundary</td><td>11</td></tr><tr><td>3. Polygons</td><td>14</td></tr><tr><td>3.1. Polygons in Ein<sup>1,n</sup></td><td>14</td></tr><tr><td>3.2. Barbot crowns</td><td>14</td></tr><tr><td>3.3. Renormalization of polygons</td><td>17</td></tr><tr><td>4. Maximal surfaces</td><td>18</td></tr><tr><td>4.1. Barbot surfaces</td><td>18</td></tr><tr><td>4.2. Renormalization</td><td>21</td></tr><tr><td>4.3. Space-horofunctions</td><td>24</td></tr><tr><td>4.4. The ideal boundary of asymptotically flat maximal surfaces</td><td>28</td></tr><tr><td>4.5. Polygonal surfaces and finite total curvature</td><td>32</td></tr><tr><td>5. The quartic differential</td><td>33</td></tr><tr><td>5.1. The quartic differential associated to maximal surfaces</td><td>33</td></tr><tr><td>5.2. Conformal type and quartic differential of an asymptotically flat maximal surface</td><td>34</td></tr><tr><td>5.3. Comparison between the space boundary and the quartic boundary</td><td>35</td></tr><tr><td>6. Main Theorem</td><td>36</td></tr><tr><td>Appendix A. The geometry of Hadamard manifolds</td><td>37</td></tr><tr><td>A.1. The Tits distance</td><td>37</td></tr><tr><td>A.2. Compact Tits boundaries and nets</td><td>39</td></tr><tr><td>A.3. Formula for the total curvature</td><td>41</td></tr><tr><td>Appendix B. Flat metrics with conical singularities</td><td>42</td></tr></table>

Date: 15 june 2024.
<table><tr><td>B.1. Ideal boundary</td><td>43</td></tr><tr><td>References</td><td>45</td></tr></table>

# 1. INTRODUCTION

## 1.1. General context.
The study of minimal surfaces is a broad and important part of differential geometry. A *minimal surface* is a critical point for the area functional on a Riemannian manifold. Some classical questions to ask are, given a boundary condition:

* Is there a minimal surface satisfying this boundary condition (the *Plateau problem*)?

* If a minimal surface exists, is it unique?

* What is the regularity one can expect for the minimal surface?

* What are the geometric properties of the surface that are imposed by the boundary condition?

When the Riemannian manifold $M$ considered has nonpositive curvature, it has an ideal boundary. We fix a simple closed curve, or a union of simple closed curves, in this boundary and ask for the existence and uniqueness of a minimal surface asymptotic to this curve. This is the *asymptotic Plateau problem*.

In the Riemannian case, already uniqueness is an issue. Surprisingly, the situation is much simpler in the pseudo-hyperbolic space $\mathbf{H}^{2,n}$.

Let us fix $(\mathbf{E}, \mathbf{q})$ a real vector space endowed with a quadratic form of non-degenerate signature $(2, n + 1)$. We define $\mathbf{H}^{2,n}$ as being the set of $\mathbf{q}$-negative definite lines of $\mathbf{E}$. It is a complete pseudo-Riemannian manifold of signature $(2, n)$ and constant curvature $-1$. The Einstein universe $\mathbf{Ein}^{1,n}$, defined as being the frontier of $\mathbf{H}^{2,n}$ in $\mathbb{P}(\mathbf{E})$, is a natural boundary for the pseudo-hyperbolic space. Given a complete spacelike surface $\Sigma$ in $\mathbf{H}^{2,n}$, we define its *space boundary* as being the set of points in $\mathbf{Ein}^{1,n}$ that are limits of sequences of points of $\Sigma$.

In [LTW22], Labourie Toulisse and Wolf study the asymptotic Plateau problem in the pseudo-hyperbolic space $\mathbf{H}^{2,n}$ for a certain class of curves. They call *semi-positive loop* in $\mathbf{Ein}^{1,n}$ a loop that is the space boundary of a complete spacelike surface of $\mathbf{H}^{2,n}$. Their result is the following: a semi-positive loop in $\mathbf{Ein}^{1,n}$ bounds a unique *maximal surface* in $\mathbf{H}^{2,n}$ (maximal surfaces are supposed to be complete in our article).

## 1.2. Main theorem.
If $\Sigma$ is a maximal surface in $\mathbf{H}^{2,n}$, then its sectional curvature takes values between $-1$ and $0$ (by [Che94]). Two extreme examples of maximal surfaces in $\mathbf{H}^{2,n}$ are the following.

* A totally geodesic copy of the hyperbolic plane, bounded by a projective circle,

* A Barbot surface, isometric to the Euclidean plane, bounded by a lightlike polygon with 4 vertices.

We study the problem of characterizing the maximal surfaces that have a behavior close to the behavior of Barbot surfaces.

Our main theorem is the following:

**Theorem 1.2.1 (Main Theorem).** Let $\Sigma$ be a maximal surface in $\mathbf{H}^{2,n}$, and denote by $g$ its induced metric. The following are equivalent:

(i) The *space boundary* of $\Sigma$ is a *lightlike polygon* with finitely many vertices,

(ii) The *total curvature* of $(\Sigma, g)$ is finite,

(iii) The *surface* $(\Sigma, g)$ is asymptotically flat.
Moreover, if the conditions are satisfied, and denoting by $N + 4$ the number of vertices of $\partial^8 \Sigma$, the Riemann surface $(\Sigma, [g])$ is of parabolic type, has polynomial quartic differential of degree $N$ and the total curvature of $\Sigma$ equals $-\frac{\pi}{2}N$.

The quartic differential is a holomorphic object associated with the second fundamental form of a maximal surface and will be defined below in Section 1.4.3.

Every maximal surface in $\mathbf{H}^{2,n}$ is a Hadamard surface. We will use techniques coming from geometry of non-positively curved manifolds to understand the geometry of maximal surfaces and its links with the space boundary.

There are three distance functions defined on a maximal surface $\Sigma$. The *induced distance function*, the *spacelike distance function*, and the *quartic distance function*.

* The induced distance function is the distance associated with the induced metric.

* The spacelike distance function between two points is the length of the $\mathbf{H}^{2,n}$-geodesic segment joining these two points.

* The quartic distance function is the distance associated with the flat metric with conical singularities coming from the quartic differential of $\Sigma$.

Remark that the first item of the Main Theorem deals with the spacelike distance function, that the second and third items deal with the induced distance function and that the fact that the quartic differential is polynomial deals with the quartic distance function.

The main idea of the proof is to assign an ideal boundary to each of the three distance functions in a natural manner, and to understand how they are related.

**1.3. Related works.** We list here some works related with ours. They deal with minimal surfaces in homogeneous spaces, and with Higgs bundles in appropriate cases.

**1.3.1. Minimal surfaces in euclidean spaces.** By a theorem of Osserman and Chern, [CO67, Theorem 1], a complete equivariant minimal surface $(\Sigma, g)$ in $\mathbb{R}^n$ has finite total curvature if and only if the following holds:

(i) the Riemann surface $X = (\Sigma, [g])$ is a compact Riemann surface with a finite number of punctures,

(ii) the minimal embedding corresponds to a collection $\{\varphi_1, \dots, \varphi_n\}$ of abelian differentials on $X$ that extend meromorphically at the punctures.

Moreover, the total curvature equals $-2\pi N$ where $N$ is a natural integer.

All finite total curvature complete minimal surfaces in the euclidean $n$-space are not classified. Some partial classification exists when we ask some topological restrictions (see [Oss97, Chapter I, Section 3]). For example, if $\Sigma$ has genus 0 in $\mathbb{R}^3$, then it is the flat plane or the catenoid.

**1.3.2. Minimal surfaces in $\mathbf{H}^2 \times \mathbb{R}$**. In [HR06], Hauswirth and Rosenberg study complete minimal surfaces in $\mathbf{H}^2 \times \mathbb{R}$ with finite total curvature. They obtain that the curvature of such surfaces must be $-2\pi N$ where $N$ is a natural integer. There are examples of such surfaces given by:

* a vertical plane, with total curvature 0,

* a minimal graph over an ideal polygon of $\mathbf{H}^2$ with $2k + 2$ vertices, $k \in \mathbb{N}^*$, and total curvature $-2\pi k$, called *Scherk graphs*.

In [PR14], Pyo and Rodríguez show that the only complete minimal surfaces with finite total curvature $-2\pi$ are the Scherk graphs with 4 vertices. They also show that the Scherk graphs are not the only examples of properly embedded minimal surfaces with finite total curvature in general.
This is the counterpart of the work of Dumas and Wolf is the setting of maximal surfaces in $H^{2,1}$.

1.3.6. Sp(4, $\mathbb{R}$) and lightlike polygons in $\mathbf{Ein}^{1,2}$. Tamburelli and Wolf [TW20] also studied polygonal maximal surfaces in $H^{2,2}$. They note that the space of polygons in $\mathbf{Ein}^{1,2}$ is not connected and describe a component of the moduli space of polygonal surfaces in $H^{2,2}$ in terms of their holomorphic data. They do not deal with the other components.

1.3.7. $G_2$ and almost complex curves in $S^{2,4}$. In another context, there is the work of Evans on polygonal surfaces in $S^{2,4}$. In [Eva24] he constructed a map from the moduli space of holomorphic polynomial sextic differentials of degree $k$ on the complex plane to the moduli space of annihilator polygons with $k + 6$ vertices in $\mathbf{Ein}^{2,3}$.

This is close to our setting. The space $S^{2,4}$ is anti-isometric to $H^{4,2}$ and the surfaces that he is considering are maximal spacelike surfaces in $H^{4,2}$.

1.3.8. Quasiperiodic surfaces in $H^{2,n}$. In [LT23], Labourie and Toulisse investigate a subset of the set of semi-positive loops in $\mathbf{Ein}^{1,n}$: the quasiperiodic loops. They found interesting characterizations concerning the maximal surfaces bounded by a quasiperiodic loop, among which the fact of having curvature bounded by above by a negative constant.

Our surfaces are, in some sense, the opposite of quasiperiodic surfaces: polygonal surfaces are asymptotically flat.

**1.4. Overview of the proof.** We explain now how goes the proof of the Main Theorem. The proof lies in a comparison between ideal boundaries associated with the three distances.

There are two main constructions for the ideal boundary of a Hadamard manifold: via rays and via horofunctions. We use both constructions to assign an ideal boundary to $\Sigma$ when endowed with these different distances.

1.4.1. *The ideal boundary.* A maximal surface $\Sigma$ in $H^{2,n}$ together with its induced metric $g$ is a Hadamard surface. The ideal boundary $\Sigma(\infty)$ of $\Sigma$ is defined as being the set of equivalence classes of unit speed geodesic rays, up to bounded distance (Section 2.3.1). The ideal boundary strongly determines the geometry of the surface via the *Tits distance* (see Appendix A). This distance is defined on the ideal boundary and by [*Oht88*], the *Tits distance* has finite diameter on $\Sigma(\infty)$ if and only if $\Sigma$ has finite total curvature.

1.4.2. *The space boundary.* We have defined the space boundary $\partial^s\Sigma$ of $\Sigma$ as being the set of limits of sequences of points in $\Sigma$ that converge in $\mathbf{Ein}^{1,n}$. As for the ideal boundary (Section 2.3.2), the space boundary can be realized as the frontier of an embedding of $\Sigma$ in the space $C^*(\Sigma) = C(\Sigma)/\mathbb{R}$ of continuous functions on $\Sigma$ modulo additive real constants.

The frontier of this embedding consists of a set of functions on $\Sigma$, defined up to an additive constant, called space-horofunctions, that are in bijection with the space boundary $\partial^s\Sigma$ and have a totally explicit description (Section 2.3.3).

**1.4.3. The quartic boundary.** We can define, using the second fundamental form on $\Sigma$, a holomorphic quartic differential on the underlying Riemann surface $(\Sigma, [g])$ as follows. We denote by $g_N$ the induced metric on the normal bundle of $\Sigma$.

The second fundamental form $\mathbf{II}$ of a maximal surface $\Sigma$ is a section of the bundle $(\mathbf{T}^*\Sigma \odot \mathbf{T}^*\Sigma) \otimes \mathcal{N}\Sigma$, where $\odot$ denotes the symmetric tensor product. Applying the
complex quadratic form $g_N^C$ to the $(2, 0)$-part of the complexified tensor $\mathbb{I}^C$ we obtain

$$ \varphi_4 := g_N^C(2\mathbb{I}^{2,0}) \in \Gamma((\mathbb{T}^{1,0}\Sigma)^{\otimes 4} \otimes \mathbb{C}) . $$

By the Codazzi equation, this is a holomorphic quartic differential on $(\Sigma, [g])$ where $[g]$ denotes the conformal class of the induced metric $g$ on $\Sigma$ (see Section 5.1).

The holomorphic quartic differential defines a flat metric with conical singularities on $\Sigma$. This metric makes $\Sigma$ a Hadamard space (in the sense of [Bal95]) and we can define the ideal boundary via rays in the same way as for a Hadamard manifold, as explained in Appendix B.

We explain now how we compare these three ideal boundaries.

### 1.4.4. Barbot surfaces and renormalization.
The simplest case of polygonal surfaces is the case of surfaces bounded by a 4-gon, called *Barbot surfaces* and *Barbot crowns*. A maximal surface $\Sigma$ has nonpositive curvature, and it is flat at a point if and only if it is flat everywhere if and only if it is a Barbot surface by [LT23, Proposition 5.3]. This is a starting point to describe asymptotically flat maximal surfaces.

An important tool is the *renormalization* process. Let $G$ be the isometry group of $\mathbf{H}^{2,n}$. The group $G$ is isomorphic to $\mathbb{PO}(2, n + 1)$. The space

$$ \{ (\Sigma, x) \mid \Sigma \text{ maximal surface, } x \in \Sigma \} $$

of pointed maximal surfaces has a natural action by $G$ that is proper, continuous and cocompact ([LTW22, Theorem 6.1] and [LT23, Theorem 4.8]). Hence, given any sequence $((\Sigma_k, x_k))_{k \in \mathbb{N}}$ of pointed maximal surfaces, there is a sequence $(g_k)_{k \in \mathbb{N}}$ of elements of $G$ such that $(g_k \cdot (\Sigma_k, x_k))_{k \in \mathbb{N}}$ subconverges to a pointed maximal surface. We say that the sequence $((\Sigma_k, x_k))_{k \in \mathbb{N}}$ subconverges *up to renormalization*.

For example, take an asymptotically flat maximal surface $\Sigma$, and a diverging sequence $(x_k)_{k \in \mathbb{N}}$ of points of $\Sigma$. Then the sequence $((\Sigma, x_k))_{k \in \mathbb{N}}$ subconverges, up to renormalization, to a Barbot surface.

We prove in Section 4.2.3 the following.

**Theorem 1.4.1.** Let $\Sigma$ be a polygonal surface in $\mathbf{H}^{2,n}$. Then for every diverging sequence $(x_k)_{k \in \mathbb{N}}$ of $\Sigma$, the sequence of pointed maximal surfaces $((\Sigma, x_k))_{k \in \mathbb{N}}$ subconverges to a pointed Barbot surface, up to renormalization.

Hence polygonal surfaces are asymptotically flat, and this is (i) ⇒ (iii) of the Main Theorem.

Barbot surfaces have an explicit description in terms of their space boundary. Also the quartic metric equals the induced metric of a Barbot surface. Hence we have a total correspondence between the space boundary, the ideal boundary and the quartic boundary of a Barbot surface.

### 1.4.5. The space boundary and the ideal boundary.
We compare here the space boundary $\partial^s \Sigma$ and the ideal boundary $\Sigma(\infty)$ of a maximal surface $\Sigma$. Recall that we identified $\partial^s \Sigma$ with the set of space-horofunctions defined on $\Sigma$ up to an additive constant.

The space-horofunctions share some properties with intrinsic horofunctions. The most important for our purpose is that space-horofunctions are strictly quasiconvex on $\Sigma$ (Theorem 4.3.2). A *space-horoball* is a sublevel set of a space-horofunction. Hence the space-horoballs are strictly convex subsets of $\Sigma$. Given a semi-positive loop $\Lambda$ in $\mathbf{H}^{2,n}$, we define

* the *vertices*, points at the intersection of two segments of photon in $\Lambda$,

* the *edges*, maximal segments of photon contained in $\Lambda$ with non-empty interior,

* the *positive points*, points meeting no edge in $\Lambda$,

* the *positive intervals*, open intervals in $\Lambda$ containing no edge.
Using a description of the shape of horoballs, we define an increasing map

$$ \Phi : \{\text{positive points}\} \cup \{\text{vertices}\} \to \Sigma(\infty). $$

**Remark 1.4.2.** The correspondence made between the space boundary and the ideal boundary is not as complete as the one in Barbot surfaces. We conjecture that the correspondence between the ideal boundary and the space boundary of polygonal surfaces is the same as in Barbot surfaces. In any case, the correspondence for vertices and positive points only is sufficient for our purpose.

Our construction is invariant by the action of $G$, hence we can apply the renormalization process to understand the Tits distance in the ideal boundary of asymptotically flat maximal surfaces. The results are the following (proved in Theorem 4.4.4 and Theorem 4.4.9):

**Theorem 1.4.3.** Let $\Sigma$ be an asymptotically flat maximal surface, and $v_1, v_2$ be adjacent vertices of $\partial^s\Sigma$. Then the Tits distance between the associated ideal points $\xi_{v_1}$ and $\xi_{v_2}$ equals $\pi/2$.

**Theorem 1.4.4.** Let $\Sigma$ be an asymptotically flat maximal surface, $I$ be a positive non-empty interval in $\partial^s\Sigma$, and $p, q$ be two distinct points of $I$. Then the Tits distance between the associated ideal points $\xi_p$ and $\xi_q$ equals $+\infty$.

As a corollary, we understand the global geometry of the metric space $(\Sigma(\infty), \mathrm{Td})$. The following is proved in Theorem 4.5.1.

**Theorem 1.4.5.** Let $\Sigma$ be a maximal surface. The Tits distance on the ideal boundary of $\Sigma$ has finite perimeter if and only if $\Sigma$ is a polygonal surface. A polygonal surface with $N + 4$ vertices has Tits perimeter equal to $N\pi/2$.

This gives the equivalence between items (i) and (ii) of the Main Theorem.

1.4.6. Asymptotically flat maximal surfaces and the quartic metric. Let now $\Sigma$ be an asymptotically flat maximal surface. Recall that the quartic metric and the induced metric are equal on Barbot surfaces. Hence for every positive number $\varepsilon$, there is a compact set $K$ in $\Sigma$ such that the induced metric $g$ and the quartic metric $g_4$ are $(1+\varepsilon)$-biLipschitz outside $K$ (Proposition 5.2.1). This makes $g_4$ a complete metric on $\Sigma$ with finitely many zeros. We prove in Theorem 5.2.4:

**Theorem 1.4.6.** If $\Sigma$ is asymptotically flat, then the Riemann surface $(\Sigma, [g])$ is biholomorphic to the complex plane and the quartic differential $\varphi_4$ is a polynomial.

We want to compare the quartic metric $g_4$ and the induced metric $g$. Since $g$ and $g_4$ are $(1+\varepsilon)$-biLipschitz outside a compact, they are $(1+\varepsilon, b)$-quasi isometric in $\Sigma$. This helps us to identify the $g$-sphere $S(r)$ and the $g_4$-sphere $S_4(r)$ of radii $r$, in a $(1+\varepsilon, b)$-quasi-isometric way. Taking the renormalized metric $d/r$ on $S(r)$ and $d_4/r$ on $S_4(r)$, and the limit when $r$ goes to $+\infty$, we obtain that the Tits boundary of $(\Sigma, g)$ and the Tits boundary of $(\Sigma, g_4)$ are isometric, yielding the following theorem (Theorem 5.3.1).

**Theorem 1.4.7.** Let $\Sigma$ be an asymptotically flat maximal surface with polynomial quartic differential of degree $N$. Then $\Sigma$ is a polygonal surface with $N + 4$ vertices.

This is (iii) ⇒ (i) of the Main Theorem, and concludes the scheme of the proof.

1.5. **Organization of the paper.** In Section 2, we recall the definitions and state some properties of $\mathbb{H}^{2,n}$ and $\mathbb{Ein}^{1,n}$. We define maximal surfaces and explain the work of Labourie Toulisse and Wolf about the asymptotic Plateau problem in $\mathbb{H}^{2,n}$. We also recall several definitions of the ideal boundary, and explain the construction of the space boundary as being the set of space-horofunctions on a maximal surface.
In Section 3 we define and study polygons in the Einstein universe. We emphasize the four vertices case, called *Barbot crowns*. In the last part of this section, we prove the *renormalization lemma*, extensively used in what follows.

In Section 4, we study the link between the space boundary $\partial^s\Sigma$ of a maximal surface $\Sigma$ and its ideal boundary $\Sigma(\infty)$. We begin with the case of Barbot surfaces, and describe the renormalization process in the case of polygonal surfaces. We study in detail space-horofunctions, allowing us to prove Theorem 4.5.1.

In Section 5, we compare the quartic metric associated with the holomorphic quartic differential and the induced metric. We prove that for an asymptotically flat maximal surface, the quartic differential is polynomial (Theorem 5.2.4) and the maximal surface is polygonal, relating the degree of the differential with the number of vertices (Theorem 5.3.1) by making a precise comparison between these two metrics.

They are two appendix at the end of the article. In Appendix A, we discuss the *Tits metric* on the ideal boundary of Hadamard surfaces. In Appendix B, we explain how the complex plane endowed with a holomorphic quartic differential can be thought of as a Hadamard space.

**Acknowledgments.** I would like to thank Parker Evans, Enrico Trebeschi and Timothé Lemistre for useful discussions about polygons and maximal surfaces in pseudo-hyperbolic spaces, and Qiongling Li for discussions about finite total curvature and Higgs bundles. I thank Andrea Seppi for having pointed Corollary 1.3.1 as a consequence of the Main Theorem. I thank my advisors Jérémy Toulisse and François Labourie for having asked me the question at the origin of this article, and for their patience and encouragements during my research and the redaction of this article.

## 2. PRELIMINARIES

Once and for all, we fix $(\mathbf{E}, \mathbf{q})$ a real quadratic space of signature $(2, n + 1)$ with a complete orientation. The polar form of $\mathbf{q}$ is denoted by $\langle, \rangle$.

We denote by

$$ \mathbb{P} : \mathbf{E} \setminus \{0\} \to \mathbb{P}(\mathbf{E}), $$

or by $[\cdot]$, the projectivization and by

$$ [\cdot]_+ : \mathbf{E} \setminus \{0\} \to \hat{\mathbb{P}}(\mathbf{E}) $$

the quotient by the action of $\mathbb{R}_+^*$ by multiplication. We denote

$$ \hat{\mathcal{G}} = \mathcal{O}(\mathbf{q}) $$

the orthogonal group of $\mathbf{q}$ and

$$ \mathcal{G} = \mathbb{P}\hat{\mathcal{G}} = \mathcal{O}(\mathbf{q}) / \{\pm I_{n+3}\} $$

its projectivized.

### 2.1. The pseudo-hyperbolic space.

#### 2.1.1. *Definition.*
The pseudo-hyperbolic space is defined as follows

$$ \mathbf{H}^{2,n} := \{x \in \mathbb{P}(\mathbf{E}) \mid \mathbf{q}|_{x} < 0\}, $$

and its double cover

$$ \hat{\mathbf{H}}^{2,n} := \{x \in \mathbf{E} \mid \mathbf{q}(x) = -1\}. $$

Both spaces are endowed with a complete pseudo-riemannian metric of signature $(2, n)$ coming from the quadratic form $\mathbf{q}$ on $\mathbf{E}$ and are of constant curvature -1.
The group $\hat{G}$ acts naturally by isometries on $\hat{\mathbf{H}}^{2,n}$ by multiplication on vectors. In the same manner, $G$ acts isometrically on $\mathbf{H}^{2,n}$. Indeed, the groups $G$ and $\hat{G}$ are the full isometry groups of $\mathbf{H}^{2,n}$ and $\hat{\mathbf{H}}^{2,n}$, respectively.

**2.1.2. Warped product structure.** An orthogonal decomposition $\mathbf{E} = P \oplus Q$ where $P$ is a positive definite plane and $Q$ is a negative definite $(n+1)$-subspace identifies $\hat{\mathbf{H}}^{2,n}$ with a warped product as follows. Denote by $\mathbf{D}^2$ the unit disc of $(P, \mathbf{q}|_P)$ and by $\mathbf{S}^n$ the unit sphere of $(Q, -\mathbf{q}|_Q)$. By [CTT19, Proposition 3.5], the map

$$ \psi : \mathbf{D}^2 \times \mathbf{S}^n \to \hat{\mathbf{H}}^{2,n} \\ (u, v) \mapsto \frac{2}{1 - \mathbf{q}(u)} u + \frac{1 + \mathbf{q}(u)}{1 - \mathbf{q}(u)} v $$

is a diffeomorphism onto and the pulled back metric $\psi^*\mathbf{q}$ equals $g_{\text{hyp}} - fg_n$, where $g_{\text{hyp}}$ is the hyperbolic metric on the Poincaré disc $\mathbf{D}^2$, $g_n$ is the round metric on the sphere $\mathbf{S}^n$ and $f : \mathbf{D}^2 \to \mathbb{R}$ is a smooth function defined by

$$ f(u) = \left( \frac{1 + \mathbf{q}(u)}{1 - \mathbf{q}(u)} \right)^2 . $$

The image by $\psi$ of $\mathbf{D}^2 \times \{v_0\}$, for $v_0$ fixed, is totally geodesic and isometric to a hyperbolic plane. It corresponds to a copy of $\hat{\mathbf{H}}^{2,0}$ because it is the quadric $\{\mathbf{q} = -1\}$ of the 3-subspace $P+\mathbb{R}v_0$ of signature $(2, 1)$. The image by $\psi$ of $\{u_0\} \times \mathbf{S}^n$, for $u_0 \in \mathbf{D}^2$, is a totally umbilic hypersurface called a *timelike sphere* and it is totally geodesic if and only if $u_0 = 0$.

The *warped projection* is, given a warped product $\psi : \mathbf{D}^2 \times \mathbf{S}^n \to \hat{\mathbf{H}}^{2,n}$, the map

$$ \pi_{\psi} : \hat{\mathbf{H}}^{2,n} \to \hat{\mathbf{H}}^{2,n} \\ p \mapsto \psi(\pi_1(\psi^{-1}(p))) $$

where $\pi_1 : \mathbf{D}^2 \times \mathbf{S}^n \to \mathbf{D}^2 \times \mathbf{S}^n$ is the projection on the first factor.

**2.1.3. Geodesics and the spacelike distance function.** Given two distinct points $x, y$ of $\mathbf{H}^{2,n}$, the set $\mathbb{P}(x \oplus y) \cap \mathbf{H}^{2,n}$ is an unparametrized geodesic. This geodesic is

* spacelike if the signature of $x \oplus y$ is $(1, 1)$,

* timelike if the signature of $x \oplus y$ is $(0, 2)$,

* lightlike if the signature of $x \oplus y$ is $(0, 1)$.

A spacelike geodesic in $\hat{\mathbf{H}}^{2,n}$ starting from a point $x$ and with direction $U$ a unit tangent vector is parametrized by arc length with the formula

$$ t \mapsto \cosh(t)x + \sinh(t)U. $$

**Definition 2.1.1 (Spacelike distance).** Let $x$ and $y$ be two different points of $\mathbf{H}^{2,n}$ on a spacelike geodesic. Take $\hat{x}$ and $\hat{y}$ lifts of $x$ and $y$ to the double cover such that $\langle \hat{x}, \hat{y} \rangle < 0$. The *spacelike distance* between $x$ and $y$ is the number

$$ \delta^s(x, y) = \arccosh(-\langle \hat{x}, \hat{y} \rangle) . $$

**Remark 2.1.2.** The definition does not depend on the choices of lifts. Moreover, given two points $x, y$ on a spacelike geodesic, the spacelike distance $\delta^s(x, y)$ equals the length of the $\mathbf{H}^{2,n}$-geodesic segment between them.

**2.1.4. The Einstein universe.** Our reference for the Einstein universe is [BCD$^+$08].

**Definition 2.1.3.** The *Einstein universe*, of signature $(1, n)$, is defined to be the projectivization of the isotropic cone of $(\mathbf{E}, \mathbf{q})$. We denote it $\mathbf{Ein}^{1,n}$. The double cover is defined as being the set of half lines in the isotropic cone of $(\mathbf{E}, \mathbf{q})$, and we denote it by $\hat{\mathbf{Ein}}^{1,n}$.
The manifold $\mathbf{Ein}^{1,n}$ is the boundary of $\mathbf{H}^{2,n}$ seen as subsets of $\mathbb{P}(\mathbf{E})$.

The Einstein universe is endowed with a conformal class of Lorentzian metrics coming from $\mathbf{q}$. The manifold $\mathbf{Ein}^{1,n}$ is always space orientable, but time orientable (hence orientable) if and only if $n$ is odd. We fix once and for all a space orientation on $\mathbf{Ein}^{1,n}$.

The group $\hat{\mathcal{G}}$ acts naturally on $\hat{\mathbf{Ein}}^{1,n}$ by multiplication on vectors $z$ belonging to the half lines $[z]_+$ in $\hat{\mathbf{Ein}}^{1,n}$. The action of $\hat{\mathcal{G}}$ is conformal. In the same manner, $\mathcal{G}$ acts conformally on $\mathbf{Ein}^{1,n}$. Indeed, the groups $\mathcal{G}$ and $\hat{\mathcal{G}}$ are the groups of conformal automorphisms of $\mathbf{Ein}^{1,n}$ and $\hat{\mathbf{Ein}}^{1,n}$, respectively.

The warped product construction extends to a product description of $\hat{\mathbf{Ein}}^{1,n}$ as being $\mathbf{S}^1 \times \mathbf{S}^n$ with the conformal class of Lorentzian metrics $[g_1 \oplus -g_n]$.

**2.1.5. Photons.** In general, two conformal metrics on a pseudo-riemannian manifold do not define the same geodesics. Even the images of the spacelike and timelike geodesics change. But for images of lightlike geodesics [GHL04, Proposition 2.131]. Lightlike geodesics are well-defined, as unparametrized curves, in $\mathbf{Ein}^{1,n}$, and we make the following definition.

**Definition 2.1.4.** A photon in $\mathbf{Ein}^{1,n}$ is a complete lightlike geodesic (seen as an unparametrized curve).

**Proposition 2.1.5.** A photon in $\mathbf{Ein}^{1,n}$ is the projectivization of an isotropic 2-plane.

*Proof.* Identify $\hat{\mathbf{Ein}}^{1,n}$ with $\mathbf{S}^1 \times \mathbf{S}^n$ by the warped product construction. Take the representative $g_1 \oplus -g_n$ of the conformal metric, and use the fact that a geodesic in a product projects to a geodesic in each factor. $\square$

**2.2. Spacelike surfaces.** By *surface* we mean an embedded and connected sub-manifold of dimension 2.

**2.2.1. Graphs.** A surface $S$ in $\hat{\mathbf{H}}^{2,n}$ is a *spacelike graph* if for any pointed hyperbolic plane, the associated warped projection restricts to a diffeomorphism between $S$ and the hyperbolic plane. We refer to [LTW22, Propositions 3.6 and 3.10] for the following proposition.

**Proposition 2.2.1.** Let $\Sigma$ be a complete spacelike surface of $\hat{\mathbf{H}}^{2,n}$. It is a spacelike graph and for any warped product decomposition $\Sigma$ is the graph of a 2-Lipschitz map from $\mathbf{D}^2$ to $\mathbf{S}^n$. A complete spacelike surface of $\mathbf{H}^{2,n}$ has two diffeomorphic lifts in $\hat{\mathbf{H}}^{2,n}$.

**2.2.2. The topology of the set of complete spacelike surfaces.** We denote by $C^\infty(\mathbf{D}^2, \mathbf{S}^n)$ the space of smooth functions from the unit disc to the $n$-sphere, endowed with the topology of uniform convergence on compact sets of all the derivatives.

By the previous Proposition, given a warped product decomposition of $\hat{\mathbf{H}}^{2,n}$, the set of complete spacelike surfaces of $\hat{\mathbf{H}}^{2,n}$ embeds in $C^\infty(\mathbf{D}^2, \mathbf{S}^n)$.

**Definition 2.2.2.** A sequence of complete spacelike surfaces $(\Sigma_k)_{k \in \mathbb{N}}$ of $\hat{\mathbf{H}}^{2,n}$ converges to a complete spacelike surface $\Sigma$ if it holds under every warped product embedding in $C^\infty(\mathbf{D}^2, \mathbf{S}^n)$. A sequence of complete spacelike surfaces of $\mathbf{H}^{2,n}$ converges if a sequence of lifts does.

**2.2.3. Semi-positive loops.** A semi-positive loop, as defined in [LTW22, Definition 2.6], is a topological circle $\gamma$ in $\mathbf{Ein}^{1,n}$ satisfying the two following properties:

* Every triple of points in $\gamma$ spans a subspace of signature different than $(1, 2)$ in $\mathbf{E}$,
• There is at least one triple of points in $\gamma$ spanning a subspace of signature (2, 1) in $\mathbf{E}$.

We use the same definition to define a semi-positive loop in $\widehat{\mathbf{Ein}}^{1,n}$.

**Proposition 2.2.3.** [LTW22, Proposition 2.11] A semi-positive loop of $\widehat{\mathbf{Ein}}^{1,n}$ is the graph of a 1-Lipschitz map from $(\mathbf{S}^1, g_1)$ to $(\mathbf{S}^n, g_n)$ in every warped product decomposition.

[LTW22, Corollary 2.13] If a semi-positive loop contains two points that lie on a photon, the loop contains a segment of photon between them.

[LTW22, Proposition 2.15] A semi-positive loop in $\widehat{\mathbf{Ein}}^{1,n}$ has two diffeomorphic lifts to the double cover. Moreover, if $\hat{p}$ and $\hat{q}$ are two points on the same lift of a semi-positive loop, then $\langle \hat{p}, \hat{q} \rangle \le 0$.

### 2.2.4. The topology of the set of semi-positive loops.

**Definition 2.2.4.** A sequence of semi-positive loops in $\widehat{\mathbf{Ein}}^{1,n}$ converges if it converges as a graph on every splitting of $\widehat{\mathbf{Ein}}^{1,n}$ as $\mathbf{S}^1 \times \mathbf{S}^n$. A sequence of semi-positive loops in $\widehat{\mathbf{Ein}}^{1,n}$ converges if a sequence of lifts does.

### 2.2.5. Asymptotic Plateau problem.
The space boundary of a submanifold $M$ of $\mathbf{H}^{2,n}$ is the subset of $\widehat{\mathbf{Ein}}^{1,n}$ defined by

$$ \partial^s M = \{ p \in \widehat{\mathbf{Ein}}^{1,n} \mid p = \lim_{k \to \infty} x_k, (x_k)_{k \in \mathbb{N}} \in M^{\mathbb{N}} \}. $$

A complete spacelike surface in $\mathbf{H}^{2,n}$ is said to be maximal if its mean curvature vector vanishes. This is equivalent to being a local maximum for the area functional [LTW22, Corollary 3.24].

In [LTW22], Labourie, Toulisse and Wolf solve the Asymptotic Plateau problem in $\mathbf{H}^{2,n}$ as follows.

**Theorem 2.2.5 (Asymptotic Plateau problem).** A semi-positive loop in $\widehat{\mathbf{Ein}}^{1,n}$ is the space boundary of a unique maximal surface in $\mathbf{H}^{2,n}$.

### 2.2.6. Compactness theorems.
A pointed maximal surface is a pair $(\Sigma, x)$ where $\Sigma$ is a maximal surface and $x$ a point of $\Sigma$. The following is [LTW22, Theorem 6.1] or [LT23, Theorem 4.8].

**Theorem 2.2.6.** The group $G$ acts properly, continuously and cocompactly on the space of pointed maximal surfaces.

We refer to [LTW22, Theorem 6.1] for the following.

**Theorem 2.2.7.** Let $(\Sigma_k)_{k \in \mathbb{N}}$ be a sequence of maximal surfaces of $\mathbf{H}^{2,n}$. For each $k$, let $\Lambda_k$ be the space boundary of $\Sigma_k$. If the sequence $(\Lambda_k)_{k \in \mathbb{N}}$ converges to a semi-positive loop $\Lambda$, then the sequence $(\Sigma_k)_{k \in \mathbb{N}}$ subconverges to the maximal surface with space boundary $\Lambda$.

### 2.3. The ideal boundary.
We explain in this section what is the ideal boundary of a Hadamard manifold, using two equivalent definitions. We mimic the second definition to show how the space boundary can be thought of as an ideal boundary for the spacelike distance function.

Recall that a *Hadamard manifold* is a simply connected, complete, non-positively curved Riemannian manifold. The reference for the material about Hadamard manifolds and their ideal boundaries is [BGS85].

#### 2.3.1. The ideal boundary via rays.
We define a *geodesic ray*, or simply a *ray*, in a complete Riemannian manifold as being a geodesic defined on $[0, +\infty)$ and parametrized by arclength.
**Definition 2.3.1.** Let $M$ be a Hadamard manifold. The ideal boundary of $M$, denoted $M(\infty)$, is defined as the quotient

$$M(\infty) := \{ \text{rays in } M \} / \sim$$

where two rays $c_1$ and $c_2$ are equivalent ($c_1 \sim c_2$) if and only if there is a real number $r$ such that $d(c_1(t), c_2(t)) < r$ for all $t \ge 0$. If $c$ is a geodesic ray, we denote by $c(\infty)$ its class in $M(\infty)$.

If $M$ is a complete Riemannian manifold and $X$ is a tangent vector of $M$ at the point $x$, then we denote by $c_X$ the ray starting at $x$ with initial velocity $X$.

**Proposition 2.3.2.** Let $M$ be a Hadamard manifold with ideal boundary $M(\infty)$. For every $\xi$ in $M(\infty)$ and every $x$ in $M$, there is a unique ray starting at $x$ representing the ideal boundary point $\xi$.

So, given any point $x$ of $M$ there is a bijection from the unit tangent bundle $T_x^1 M$ to the ideal boundary $M(\infty)$. We can give a topology to $M(\infty)$ via this bijection. It turns out that this topology does not depend on the chosen point $x$.

An ideal boundary point $\xi$ of a Hadamard manifold can be seen as a collection of rays, one for each point of the manifold. We denote by $\xi_x$ the representative of an ideal boundary point $\xi$ at a point $x$.

**2.3.2. The ideal boundary via horofunctions.** Another equivalent way to define the ideal boundary is via horofunctions. Let $M$ be a complete Riemannian manifold. Denote by $C(M)$ the space of continuous functions on $M$, with the topology of uniform convergence on compact sets, that makes it a complete space. Denote by $C^*(M)$ the quotient of this space by the equivalence relation $f \sim g$ if $f = g + r$ for $r$ a real number. Then $M$ embeds in $C^*(M)$ via distance functions:

$$\psi : M \to C^*(M) \\ x \mapsto [d(x, \cdot)].$$

**Definition 2.3.3.** A horofunction on $M$ is an element of the frontier of $\psi(M)$ in $C^*(M)$. The horo boundary of $M$, denoted $\partial^h M$, is the set of horofunctions.

It is shown in [BGS85, 3.6] that, if $M$ is a Hadamard manifold, the ideal boundary $M(\infty)$ and the horo boundary $\partial^h M$ are homeomorphic, using the following identification.

Given an ideal boundary point $\xi$ in $M(\infty)$, we can look at a limit of renormalized distance functions as follows: take a base point $x_0$ in $M$, denote by $c$ the ray starting at $x_0$ with direction $\xi$, and define the curve

$$[0, +\infty) \to C(M) \\ t \mapsto d(c(t), \cdot) - d(c(t), x_0).$$

This curve has a limit when $t$ goes to $+\infty$, and this limit cannot be a distance function, hence it is a horofunction. One can show that the construction of the limit function does not depend on the choice of the base point, but only on the direction $\xi$. We call these functions Busemann functions. It turns out that, conversely, all horofunctions are Busemann functions for a certain ideal boundary point.

**2.3.3. The space boundary of complete spacelike surfaces.** Let $\Sigma$ be a complete spacelike surface of $H^{2,n}$. Recall that its space boundary $\partial^s \Sigma$ is defined as being the subset of $\text{Ein}^{1,n}$ containing the limits of diverging sequences in $\Sigma$. We now present another point of view, imitating the second construction of the ideal boundary. As in the previous section, $C(\Sigma)$ is the space of continuous functions on $\Sigma$ endowed with the topology of uniform convergence on compact sets and $C^*(\Sigma)$ is the quotient of this space by constant functions.
We embed $\Sigma$ in the space $C^*(\Sigma)$ via spacelike distance functions $\delta^s$ (recall Definition 2.1.1):

$$ \psi^s : \Sigma \to C^*(\Sigma) \\ x \mapsto [\delta^s(x, \cdot)] . $$

As for the ideal boundary, we want to characterize the frontier of this embedding. Before making computations, we recall that a complete spacelike surface of $\mathbf{H}^{2,n}$ admits two diffeomorphic lifts to the double cover (Proposition 2.2.3).

**Definition 2.3.4.** The space-horofunction associated to the point $p$ of $\widehat{\mathbf{Ein}}^{1,n}$ is the family of functions defined on $\widehat{\mathbf{H}}^{2,n} \setminus L(p)$, up to an additive constant, by

$$ h_p(x) = \log(|\langle z, \hat{x} \rangle|), $$

for $z$ in $p$.

Here $L(p)$ denotes the lightcone of $p$, that is the set of points of $\widehat{\mathbf{H}}^{2,n}$ that are in lightlike geodesics of $\widehat{\mathbf{H}}^{2,n}$ asymptotic to $p$. Remark that, such defined, $h_p = h_{-p}$, and $h_p(-x) = h_p(x)$ for any $p$ in $\widehat{\mathbf{Ein}}^{1,n}$ and $x$ in $\widehat{\mathbf{H}}^{2,n}$. Hence we also define the space-horofunction associated to a point $p$ in $\widehat{\mathbf{Ein}}^{1,n}$, and denote it by $h_p$, with the formula

$$ h_p : \widehat{\mathbf{H}}^{2,n} \setminus L(p) \to \mathbb{R}, \quad x \mapsto h_{\hat{p}}(\hat{x}), $$

where $\hat{p}$ and $\hat{x}$ are lifts of $p$ and $x$ to the double covers $\widehat{\mathbf{Ein}}^{1,n}$ and $\widehat{\mathbf{H}}^{2,n}$, respectively.

**Theorem 2.3.5.** The frontier of $\psi^s(\Sigma)$ in $C^*(\Sigma)$ is the set

$$ \{h_p \mid p \in \partial^s \Sigma\} $$

of space-horofunctions associated with points of $\partial^s \Sigma$. This defines a homeomorphism $\Sigma \cup \partial^s \Sigma \to \bar{\psi}^s(\Sigma)$.

We make the computations in the double cover.

*Proof.* The set $\{h_p \mid p \in \partial^s \Sigma\}$ is contained in the frontier of $\psi^s(\Sigma)$ as follows.

Let $p$ be a point in $\partial^s \Sigma$ and denote by $h_p$ the associated space-horofunction. Let $(x_k)_{k \in \mathbb{N}}$ be a sequence of points in $\Sigma$ that tends to $p$. Take a vector $z$ in $p$ and $\hat{x}_k$ a lift of $x_k$ to the double cover such that $\langle \hat{x}_k, z \rangle$ is negative for each integer $k$. Take a sequence $(\lambda_k)_{k \in \mathbb{N}}$ of positive real numbers such that $(\lambda_k \hat{x}_k)_{k \in \mathbb{N}}$ tends to $z$. Since the $\hat{x}_k$'s are in $\{\mathbf{q} = -1\}$, the sequence $(\lambda_k)_{k \in \mathbb{N}}$ tends to 0. By definition we have

$$ \delta^s(x_k, x) - \delta^s(x_k, x_0) = \operatorname{arccosh}(-\langle \hat{x}_k, \hat{x} \rangle) - \operatorname{arccosh}(-\langle \hat{x}_k, \hat{x}_0 \rangle). $$

Since $\operatorname{arccosh}(t) = \log(t + \sqrt{t^2 - 1})$ for every $t \ge 1$, we have

$$ \begin{aligned} \exp(\delta^s(x_k, x) - \delta^s(x_k, x_0)) &= \frac{-\langle \hat{x}_k, \hat{x} \rangle + \sqrt{\langle \hat{x}_k, \hat{x} \rangle^2 - 1}}{-\langle \hat{x}_k, \hat{x}_0 \rangle + \sqrt{\langle \hat{x}_k, \hat{x}_0 \rangle^2 - 1}} \\ &= \frac{\langle \lambda_k \hat{x}_k, \hat{x} \rangle - \sqrt{\langle \lambda_k \hat{x}_k, \hat{x} \rangle^2 - \lambda_k^2}}{\langle \lambda_k \hat{x}_k, \hat{x}_0 \rangle - \sqrt{\langle \lambda_k \hat{x}_k, \hat{x}_0 \rangle^2 - \lambda_k^2}}, \end{aligned} $$

and this function converges uniformly on the compact sets to the function

$$ x \mapsto \frac{\langle z, \hat{x} \rangle}{\langle z, \hat{x}_0 \rangle}, $$

when $k \to +\infty$. We extend $\psi^s$ to $\partial^s \Sigma$ by $\psi^s(p) = h_p$.

Now if $p, q$ are two distinct points of $\partial^s \Sigma$, the associated space-horofunctions are distinct in $C^*(\Sigma)$. For the last assertion, since $\delta^s(x, \cdot)$ and $h_p$ are explicit, a computation as above shows that for every sequence $(x_k)_{k \in \mathbb{N}}$ in $\Sigma \cup \partial^s \Sigma$ and $x$ in $\Sigma \cup \partial^s \Sigma$, the sequence $(x_k)_{k \in \mathbb{N}}$ converges to $x$ if and only if the sequence $(\psi^s(x_k))_{k \in \mathbb{N}}$ converges to $\psi^s(x)$. $\square$
**Remark 2.3.6.** This construction gives an extrinsic ideal boundary, even if the ambient space is not Riemannian. It can be generalized in various pseudo-Riemannian manifolds where complete non-compact acausal submanifolds can be endowed with a space boundary in the same manner.

### 3. POLYGONS

The aim of this section is to define the *lightlike polygons* in the Einstein universe, and study some of their properties.

**3.1. Polygons in $\mathbf{Ein}^{1,n}$**. The notion of segment we have in $\mathbf{Ein}^{1,n}$ is that of segment of photons. Recall (Definition 2.1.4 and Proposition 2.1.5) that a photon is the image of a complete lightlike geodesic and is indeed the projectivization of an isotropic plane of $\mathbf{E}$.

**Definition 3.1.1 (Lightlike polygons).** A *lightlike polygon* in $\mathbf{Ein}^{1,n}$ is a semi-positive loop consisting of finitely many segments of photons. We write a $N$-gon to designate a lightlike polygon with $N$ vertices.

**Definition 3.1.2.** Let $\Lambda$ be a semi-positive loop in $\mathbf{Ein}^{1,n}$. A *vertex* is a point of $\Lambda$ having a neighborhood, in $\Lambda$, consisting of the union of two segments of different photons.

**Definition 3.1.3.** Let $\Lambda$ be a semi-positive loop in $\mathbf{Ein}^{1,n}$. An *edge* of $\Lambda$ is a maximal segment of photon included in $\Lambda$ and containing at least two points.

**Proposition 3.1.4.** Every lightlike polygon has at least four vertices.

*Proof.* Given two points $p, q$ in $\mathbf{Ein}^{1,n}$, there is at most 1 photon passing through them. Hence there is no 2-gon. The vertices of a 3-gon span a 3-dimensional totally isotropic plane in $\mathbf{E}$ and this is impossible. $\square$

**Remark 3.1.5.** The 2-gons exist in $\widehat{\mathbf{Ein}}^{1,1}$. Take $p$ in $\widehat{\mathbf{Ein}}^{1,1}$ and $q = -p$. There are four segment of photons between $p$ and $q$. Indeed there are two photons $\phi$ and $\psi$ passing through $[p]$ in $\widehat{\mathbf{Ein}}^{1,1}$ hence four lifts $\phi_+, \phi_-, \psi_+, \psi_-$ in the double cover. If $\phi_+$ and $\psi_+$ contain points with positive scalar product, then $\phi_+ \cup \psi_-$ is a 2-gon. Otherwise $\phi_+ \cup \psi_+$ is a 2-gon. Every 2-gon in $\widehat{\mathbf{Ein}}^{1,n}$ is of this form. The projection of the 2-gon in $\widehat{\mathbf{Ein}}^{1,1}$ is the union of the two photons $\phi \cup \psi$.

The first interesting case is the object of the next section.

**3.2. Barbot crowns.** We study here the case of 4-gons, called *Barbot crowns* in [LTW22]. A Barbot crown is always included inside a copy of $\widehat{\mathbf{Ein}}^{1,1}$. Indeed, take a Barbot crown $C$ with vertices $v_1, v_2, v_3, v_4$. Take a lift of $C$ in $\widehat{\mathbf{Ein}}^{1,n}$ with vertices $\hat{v}_i$, and let $z_i$ in $\hat{v}_i$ be non zero vectors. Two neighboring vertices lie on a photon, so $\langle z_i, z_{i+1} \rangle = 0$. Two vertices that are not adjacent must be transverse for at least two reasons:

* if not, then we have three points spanning a totally isotropic 3-plane, it is impossible in $\mathbf{E}$,

* if two points of a semi-positive loop lie on a photon, the loop must contain a segment of photon between them (Proposition 2.2.3).

Hence, we have

$$ \langle\langle z_i, z_j \rangle\rangle_{1 \le i, j \le 4} = \begin{pmatrix} 0 & 0 & \langle z_1, z_3 \rangle & 0 \\ 0 & 0 & 0 & \langle z_2, z_4 \rangle \\ \langle z_1, z_3 \rangle & 0 & 0 & 0 \\ 0 & \langle z_2, z_4 \rangle & 0 & 0 \end{pmatrix}, $$
with determinant equal to $\langle z_1, z_3 \rangle^2 \langle z_2, z_4 \rangle^2 > 0$. At the end, the vectors $\{z_i\}$ span a non-degenerate subspace of $\mathbf{E}$, of positive determinant and containing totally isotropic planes. Hence $v_1 \oplus v_2 \oplus v_3 \oplus v_4$ is of signature $(2,2)$ and $C$ is included in the projectivization $\mathbb{P}(v_1 \oplus v_2 \oplus v_3 \oplus v_4) \cap \mathbf{Ein}^{1,n} \simeq \mathbf{Ein}^{1,1}$.

**Definition 3.2.1.** We say that a quadruple $(z_1, z_2, z_3, z_4)$ of distinct isotropic vectors of $\mathbf{E}$ is a hyperbolic basis if

* $\mathbb{R}z_i \oplus \mathbb{R}z_{i+1}$ is a photon for each $1 \le i \le 4$,

* $\langle z_i, z_{i+2} \rangle = -1/4$ for $i=1$ and $i=2$.

**Proposition 3.2.2.** Given $v_1, v_2, v_3, v_4$ vertices of a Barbot crown, there are $z_1, z_2, z_3, z_4$ in $v_1, v_2, v_3, v_4$ respectively that form a hyperbolic basis.

*Proof.* Take $w_1, w_2, w_3, w_4$ non zero vectors in $v_1, v_2, v_3, v_4$, respectively. The first property is automatically satisfied and $\langle w_i, w_{i+2} \rangle \ne 0$. The vectors

$$z_1 = w_1, z_2 = w_2, z_3 = -\frac{1}{4\langle w_1, w_3 \rangle}w_3 \text{ and } z_4 = -\frac{1}{4\langle w_2, w_4 \rangle}w_4$$

then form a hyperbolic basis.

<figure><img src="image_3.png" /></figure>

**Proposition 3.2.3.** A hyperbolic basis $(z_i)_{1 \le i \le 4}$ determines in a unique way a Barbot crown in $\widehat{\mathbf{Ein}}^{1,n}$ passing through the vertices $\hat{v}_i = \hat{\pi}(z_i)$, hence a unique Barbot crown in $\mathbf{Ein}^{1,n}$ passing through the vertices $v_i = \pi(z_i)$.

*Proof.* For each $i$ in $\mathbb{Z}/4\mathbb{Z}$, the segment $\{(1-t)z_i + tz_{i+1} \mid t \in [0,1]\}$ gives rise to a segment of photon between $\hat{v}_i$ and $\hat{v}_{i+1}$, in $\widehat{\mathbf{Ein}}^{1,n}$. The union of these four segments is a topological circle $\hat{\Lambda}$, and is semi-positive. Indeed, the inequality

$$\langle (1 - t)z_i + tz_{i+1}, (1 - s)z_j + sz_{j+1} \rangle \le 0,$$

for all $s,t$ in $(0,1)$ and $i,j$ in $\{1, \dots, 4\}$ proves that triples of points of $\Lambda$ cannot span a space of signature $(1,2)$ and that a triple of $\Lambda$ not containing two points on a photon span a space of signature $(2,1)$. Hence we have defined a Barbot crown in $\widehat{\mathbf{Ein}}^{1,n}$, that projects to a Barbot crown in $\mathbf{Ein}^{1,n}$. The obtained Barbot crown in $\widehat{\mathbf{Ein}}^{1,n}$ is unique. Indeed, we have $\langle z_i, (1-t)z_j + tz_{j+1} \rangle > 0$ for $\{t < 0 \text{ and } i = j-1\}$ and for $\{t > 1 \text{ and } i = j+2\}$ (recall that two points $x,y$ on a semi-positive loop must satisfy $\langle x,y \rangle \le 0$ by Proposition 2.2.3). $\square$

<figure><img src="image_4.png" /></figure>

**Proposition 3.2.4.** The group $\hat{\mathcal{G}}$ acts transitively on the set of hyperbolic basis. The group $\mathcal{G}$ acts transitively on the set of Barbot crowns.

*Proof.* Fix a basis $e_1, \dots, e_{n+1}$ of $\mathbf{E}$ where $\mathbf{q}$ is written

$$\mathbf{q}(x) = -x^1 x^3 - x^2 x^4 - \sum_{5 \le i \le n+3} (x^i)^2,$$

for $x = \sum_i x^i e_i$. Let $(z_1, z_2, z_3, z_4)$ be a hyperbolic basis. The map sending each $z_i$ to $\frac{1}{2}e_i$ respects the quadratic form $\mathbf{q}$ and can therefore be extended to an isometry of $\mathbf{E}$, giving an element of $\hat{\mathcal{G}}$ sending the hyperbolic basis to $(\frac{1}{2}e_1, \frac{1}{2}e_2, \frac{1}{2}e_3, \frac{1}{2}e_4)$.

Let $v_1, v_2, v_3, v_4$ be the ordered vertices of a Barbot crown $C$ in $\mathbf{Ein}^{1,n}$. Take a lift $\hat{C}$ of $C$ to the double cover $\widehat{\mathbf{Ein}}^{1,n}$ with $z_i$ in $\hat{v}_i$ non-zero vectors. Then, up to the multiplication of each $z_i$ by a positive constant, $(z_1, z_2, z_3, z_4)$ is an hyperbolic basis. Since $\hat{\mathcal{G}}$ acts transitively on the set of hyperbolic basis, we obtain the result. $\square$

<figure><img src="image_5.png" /></figure>
### 3.2.1. Determination of a Barbot crown.
We describe here a way to determine a unique Barbot crown, given three points in a certain configuration.

**Proposition 3.2.5.** Given three points $v_1, v_2, v_3$ of $\text{Ein}^{1,n}$ satisfying the following properties:

* $v_1 \oplus v_2$ and $v_2 \oplus v_3$ are photons,

* $v_1 \oplus v_3$ is of signature $(1, 1)$ for $\mathbf{q}$,

there is a point $v_4$ in $\text{Ein}^{1,n}$ such that the points $v_i$ are the vertices of a Barbot crown.

If $n$ equals 1, the point $v_4$ is unique. If $n$ is bigger than 1, then the point is not unique, there is a $n-1$-dimensional manifold of possible choices.

*Proof.* We search a vector in $v_1^{\perp} \cap v_3^{\perp}$ transverse to $v_2$. The subspace

$$v_1^{\perp} \cap v_3^{\perp} = (v_1 \oplus v_3)^{\perp}$$

is of dimension $n+1$ and signature $(1, n)$ for $\mathbf{q}$, and contains $v_2$. Hence the result.

□

**Corollary 3.2.6.** Let $\Lambda$ be a semi-positive loop containing a vertex $v$. Then there is a Barbot crown that coincides with $\Lambda$ along the edges passing through $v$.

*Proof.* We denote by $v_2$ the point $v$ and by $v_1, v_3$ the two points at the end of the edges, in $\Lambda$, passing through $v_2$. Then we can apply Proposition 3.2.5 so that there is $v_4$ in $\text{Ein}^{1,n}$ such that the points $v_i$ are the vertices of a Barbot crown. Since there are two Barbot crowns passing through the points $v_i$ it remains to show that one of them works.

Take a lift of $\Lambda$ to the double cover. Because of the condition 2.2.3, one of the two Barbot crowns works. □

### 3.2.2. Barbot crowns and Cartan subspaces.
Take a Barbot crown $C$ whose vertices are $v_1, v_2, v_3, v_4$. Take $z_i$ in $v_i$ for each $1 \le i \le 4$ such that $(z_1, z_2, z_3, z_4)$ is a hyperbolic basis. Define $\mathcal{A}$ as being the subgroup of $\hat{\mathcal{G}}$ consisting of elements stabilizing each vertex $v_i$ and stabilizing the orthogonal complement $(v_1 \oplus \dots \oplus v_4)^{\perp}$ point wise. The group $\mathcal{A}$ can also be seen as a subgroup of $\mathcal{G}$. Let $(e_5, \dots, e_{n+3})$ be an orthonormal basis of $(v_1 \oplus v_2 \oplus v_3 \oplus v_4)^{\perp}$. Hence $(z_1, \dots, z_4, e_5, \dots, e_{n+3})$ is a basis of $\mathcal{E}$ and we write elements of $\hat{\mathcal{G}}$ with respect to this basis.

**Proposition 3.2.7.** The identity component of the subgroup $\mathcal{A}$ is the exponential of a Cartan subspace of $\mathfrak{g}$.

*Proof.* Indeed,

$$A_0 = \left\{ \begin{pmatrix} \lambda & & & \\ & \mu & & \\ & & \lambda^{-1} & \\ & & & \mu^{-1} \\ & & & I_{(n-1)} \end{pmatrix} \middle| \lambda, \mu \in \mathbb{R}_+^* \right\}.$$

The Lie algebra of $\mathcal{A}$ is then

$$\mathfrak{a} = \left\{ \begin{pmatrix} u & & & \\ & v & & \\ & & -u & \\ & & & -v \\ & & & 0_{(n-1) \times (n-1)} \end{pmatrix} \mid u, v \in \mathbb{R} \right\},$$

and one can show that $\mathfrak{a}$ is a Cartan subspace of $\mathfrak{g}$. □
We call A the Cartan subgroup associated to C and denote its elements by

$$ a(\lambda, \mu) := \begin{pmatrix} \lambda & & & \\ & \mu & & \\ & & \lambda^{-1} & \\ & & \mu^{-1} & I_{(n-1)} \end{pmatrix}, $$

for $\lambda, \mu$ real numbers.

**3.3. Renormalization of polygons.** We prove a lemma that we use to renormalize a class of semi-positive loops into Barbot crowns. Let $C$ be a Barbot crown with vertices $v_1, v_2, v_3, v_4$. Denote by $\phi$ the edge between $v_1$ and $v_2$ in $C$. Denote by $\psi$ the edge between $v_2$ and $v_3$ in $C$.

**Lemma 3.3.1 (Renormalization lemma near a vertex).** Let $\Lambda$ be a semi-positive loop having $\phi$ and $\psi$ as edges. Let $(a_k)_{k \in \mathbb{N}}$ be a sequence of $A_0$, the Cartan subgroup associated to $C$, where $a_k = a(\lambda_k, \mu_k)$ satisfies the conditions

(1) the sequence $(\mu_k)_{k \in \mathbb{N}}$ tends to 0,

(2) the sequence $(\lambda_k \mu_k)_{k \in \mathbb{N}}$ is bounded by above,

(3) the sequence $(\lambda_k^{-1} \mu_k)_{k \in \mathbb{N}}$ is bounded by above.

Then the sequence of semi-positive loops $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ converges to $C$.

*Proof.* Take $z_1, z_2, z_3, z_4$ in $v_1, v_2, v_3, v_4$ respectively such that $(z_1, z_2, z_3, z_4)$ is a hyperbolic basis. Now write every point $p$ of $\text{Ein}^{1,n}$ as

$$ p = \mathbb{P}(p_1 z_1 + p_2 z_2 + p_3 z_3 + p_4 z_4 + p_5), $$

for $p_i$ in $\mathbb{R}$ and $p_5$ in $(v_1 \oplus v_2 \oplus v_3 \oplus v_4)^\perp \simeq \mathbb{R}^{0,n-1}$. We use the notation $p = [p_1 : p_2 : p_3 : p_4 : p_5]$. Recall that the action of an element $a(\lambda, \mu)$ on the point $p$ is

$$ a(\lambda, \mu) \cdot p = [\lambda p_1 : \mu p_2 : \lambda^{-1} p_3 : \mu^{-1} p_4 : p_5]. $$

There are several possibilities for the sequences $(\lambda_k \mu_k)_{k \in \mathbb{N}}$. We want to show that every subsequence of $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ admits a subsequence tending to $C$. We will show that a sublimit of $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ must contain $v_1, v_2, v_3$ and $v_4$. Remark that, up to extracting a subsequence, $\lambda_k \ge 1$ or $\lambda_k \le 1$. The two cases are similar so we only consider the case $\lambda_k \ge 1$. In our case, the sequence $(\lambda_k^{-1} \mu_k)_{k \in \mathbb{N}}$ tends to 0.

• Suppose that $(\lambda_k \mu_k)_{k \in \mathbb{N}}$ tends to 0. Take any point $p$ in $\Lambda \setminus (\phi \cup \psi)$. By assumptions, it is not on an edge with $v_2$, hence

$$ p = [p_1 : p_2 : p_3 : 1 : p_5], $$

for $p_1, p_2, p_3$ in $\mathbb{R}$ and $p_5$ in $(v_1 \oplus v_2 \oplus v_3 \oplus v_4)^\perp$. Acting with $a_k$ we obtain

$$ a_k \cdot p = [\lambda_k p_1 : \mu_k p_2 : \lambda_k^{-1} p_3 : \mu_k^{-1} : p_5] = [\lambda_k \mu_k p_1 : \mu_k^2 p_2 : \frac{\mu_k}{\lambda_k} p_3 : 1 : \mu_k p_5], $$

and it tends to $v_4$ since $(\mu_k)_{k \in \mathbb{N}}$, $(\lambda_k \mu_k)_{k \in \mathbb{N}}$ and $(\lambda_k^{-1} \mu_k)_{k \in \mathbb{N}}$ tend to 0.

• Suppose now that $(\lambda_k \mu_k)_{k \in \mathbb{N}}$ is bounded from below. Then $(\lambda_k)_{k \in \mathbb{N}}$ tends to $+\infty$. Up to extraction, the sequence $(\lambda_k \mu_k)_{k \in \mathbb{N}}$ tends to a positive real number $r$. Take now $p$ in $\Lambda \setminus (\phi \cup \psi)$. If $p$ is in an edge with $v_3$ then $p_1 = 0$. By the same computation as above we conclude that $(a_k \cdot p)_{k \in \mathbb{N}}$ tends to $v_4$. It may not exist such a $p$ in $\Lambda \setminus (\phi \cup \psi)$. In this case, take $p$ in $\Lambda \setminus (\phi \cup \psi)$ that is neither in an edge with $v_3$ neither in an edge with $v_1$ and write

$$ p = [p_1 : p_2 : 1 : p_4 : p_5], $$

with $p_1, p_4 \ne 0$ ($p$ is not in an edge with $v_2$ either). The norm of the underlying vector is zero, that is

$$ -\frac{1}{2}(p_1 + p_2 p_4) + \mathbf{q}(p_5) = 0. $$
Since $p_5$ lies in a negative definite space for $\mathbf{q}$, we have

$$p_1 + p_2p_4 \le 0,$$

hence

$$(1) \qquad \frac{p_1}{p_4} \le -p_2.$$

Now the vector $\hat{p} = p_1z_1 + p_2z_2 + z_3 + p_4z_4 + p_5$ satisfies

$$\langle \hat{p}, z_3 \rangle < 0 \text{ and } \langle \hat{p}, z_2 \rangle < 0.$$

Indeed $\Lambda$ is a semi-positive loop and one lift of $C$ has vertices spanned by $z_1, z_2, z_3$ and $z_4$ (recall Proposition 2.2.3) and we already have $\langle p, z_1 \rangle = -1/4 < 0$. So we have

$$p_1 > 0 \text{ and } p_4 > 0.$$

By equation (1), we obtain that $p_2 < 0$. Now by taking the point $p$ close to $v_3$ in $\Lambda \setminus (\phi \cup \psi)$, we can suppose that $|p_1|, |p_2|, |p_4|$ and $\mathbf{q}(p_5)$ are as close to 0 as we want. Again by equation (1), $p_1/p_4$ is as close to 0 as we want. Now, looking at the sequence $(a_k \cdot p)_{k \in \mathbb{N}}$, we have

$$\lim_{k \to +\infty} a_k \cdot p = \lim_{k \to +\infty} \left[ \frac{\lambda_k \mu_k p_1}{p_4} : \frac{\mu_k^2 p_2}{p_4} : \frac{\mu_k p_3}{\lambda_k p_4} : 1 : \frac{\mu_k p_5}{p_4} \right] = \left[ r \frac{p_1}{p_4} : 0 : 0 : 1 : 0 \right].$$

Hence, if the limit of the sequence $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ exists, it contains points as close to $v_4$ as we want, hence it must contain $v_4$.

Now, in every case, the sequence $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ subconverges either to a photon or a semi-positive loop, by [LT23, Corollary 2.25]. Up to extracting a subsequence, $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ converges to a semi-positive loop containing $v_1, v_2, v_3$ (because each isometry $a_k$ preserves these points) and $v_4$ (by what precedes). Since a semi-positive loop contains a photon segment between two points as soon as two points lie on such a segment (Proposition 2.2.3), any sublimit of $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ is a Barbot crown with vertices $v_1, v_2, v_3$ and $v_4$. But since each $a_k$ preserves the segment between $v_i$ and $v_{i+1}$, for $1 \le i \le 2$, every sublimit of $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ is $C$. $\square$

## 4. MAXIMAL SURFACES

In this section, we study the link between the ideal boundary of a maximal surface $\Sigma$ and its space boundary. As we have seen in Section 2.3.3, the space boundary $\partial^s\Sigma$ can be identified with a set of space-horofunctions, allowing us to compare it with the ideal boundary $\Sigma(\infty)$.

We start by a study of Barbot surfaces in detail in Section 4.1. In Section 4.2 we define a projection between maximal surfaces. Then, in Section 4.3, we study our most important tool, the *space-horofunctions* and their properties. With this toolbox, we compute the *Tits distance* of the ideal boundary of asymptotically flat maximal surfaces in Section 4.4 and link the total curvature with $\partial^s\Sigma$ in Section 4.5.

### 4.1. Barbot surfaces.
A Barbot surface is a maximal surface having a Barbot crown as space boundary. The Barbot surfaces are the easiest polygonal surfaces to study: they have the least number of vertices (Proposition 3.1.4). These surfaces have flat induced metric and a totally explicit description. We begin with a description of Barbot surfaces, given the four vertices of their space boundary. We also describe the geodesics of Barbot surfaces, and in particular their limit points in $\text{Ein}^{1,n}$. At the end of this section we discuss space-horofunctions in Barbot surfaces.
4.1.1. *Explicit parametrization.* Let $v_1, v_2, v_3, v_4$ be the vertices of a Barbot crown $C$. Let $S$ be the maximal surface with space boundary $C$. Recall that the Cartan subgroup $\mathcal{A}$ in $\mathcal{G}$ corresponding to $C$ is defined to be the subgroup stabilizing the four vertices $v_i$ globally and $(v_1 \oplus v_2 \oplus v_3 \oplus v_4)^\perp$ pointwise. Let $(z_1, z_2, z_3, z_4)$ be a hyperbolic basis with $z_i$ in $v_i$ and define $x = [z_1 + z_2 + z_3 + z_4]$.

**Proposition 4.1.1.** *The Barbot surface S is the orbit of x under the action of A₀ and has flat induced metric.*

*Proof.* The orbit $\mathcal{A}_0 \cdot x$ has vanishing mean curvature at $x$, hence it is a maximal surface by homogeneity. Since $\mathcal{A}_0 \cdot x$ is included in $v_1 \oplus v_2 \oplus v_3 \oplus v_4$, we consider only the restriction of $\mathcal{A}$ to $v_1 \oplus v_2 \oplus v_3 \oplus v_4$. We do the computations in the double cover, where $[z_i]_+$ are the vertices of a lift of $C$ and where the lift of $x$ is still denoted $x$. One parameter subgroups of $\mathcal{A}_0$ are given by

$$ \gamma_X(t) = \begin{pmatrix} \exp(tu) & 0 & 0 & 0 \\ 0 & \exp(tv) & 0 & 0 \\ 0 & 0 & \exp(-tu) & 0 \\ 0 & 0 & 0 & \exp(-tv) \end{pmatrix}, $$

in the basis $(z_1, z_2, z_3, z_4)$ and with $u, v$ in $\mathbb{R}$. We identify $\mathfrak{a} = T_x(\mathcal{A}_0 \cdot x)$ with $\mathbb{R}^2$ via the isometric map that sends the matrix of $\mathfrak{a}$ with diagonal $(u, v, -u, -v)$ to the vector $(u, v)$. Hence a geodesic ray $\sigma_X$ of $\mathcal{A}_0 \cdot x$ starting from $x$ with initial velocity $X = (u, v)$ is parametrized as follows:

$$ (2) \quad \sigma_X(t) = \gamma_X(t) \cdot x = \exp(tu)z_1 + \exp(tv)z_2 + \exp(-tu)z_3 + \exp(-tv)z_4. $$

Now taking orthogonal unit tangent vectors $X = (1, 0)$ and $Y = (0, 1)$, the mean curvature vector $H$ of $\mathcal{A}_0 \cdot x$ is given by

$$ H(x) = \mathbb{I}(\dot{\sigma}_X(0), \dot{\sigma}_X(0)) + \mathbb{I}(\dot{\sigma}_Y(0), \dot{\sigma}_Y(0)) = (z_1 + z_3 + z_2 + z_4)^\perp = 0. $$

In the last equality, we take the part of the vector normal to $T_x(\mathcal{A}_0 \cdot x)$. So $\mathcal{A}_0 \cdot x$ is a maximal surface. It is flat and complete, being the orbit of the isometric action of $\mathcal{A}_0 \simeq \mathbb{R}^2$. It remains to show that the space boundary of $\mathcal{A}_0 \cdot x$ is $\hat{C}$. One can verify this by looking at limits of intrinsic geodesics of $\mathcal{A}_0 \cdot x$. $\square$

We can parametrize, given $(z_1, z_2, z_3, z_4)$ a hyperbolic basis, a Barbot surface $S$ associated to this hyperbolic basis by

$$ f: \mathbb{R}^2 \to \hat{\mathbf{H}}^{2,n} \\ (u, v) \mapsto \exp(u)z_1 + \exp(v)z_2 + \exp(-u)z_3 + \exp(-v)z_4. $$

We can now describe the tangent and normal bundle of the Barbot surface $S$. Denote by $\mathbf{H}^{2,1}$ the projectivization of the set of negative lines of $v_1 \oplus \dots \oplus v_4$. Let us define

* $\partial_u = \exp(u)z_1 - \exp(-u)z_3 = f_* \frac{\partial}{\partial u}$,

* $\partial_v = \exp(v)z_2 - \exp(-v)z_4 = f_* \frac{\partial}{\partial v}$,

* $\mathbf{n} = \exp(u)z_1 - \exp(v)z_2 + \exp(-u)z_3 - \exp(-v)z_4$.

**Proposition 4.1.2** (Description of the tangent and normal bundles of a Barbot surface). At a point $x = f(u, v)$, the tangent bundle of the Barbot surface $S$ is spanned by the vectors $\partial_u$ and $\partial_v$. The normal bundle of $S$ splits in two parts, one in the tangent space of $\mathbf{H}^{2,1}$ spanned by the vector $\mathbf{n}$ and the other one in $T\mathbf{H}^{2,1}\perp$. Moreover, the second fundamental form $\mathbb{I}$ of $S$ satisfies, at the point $x = f(u, v)$,

$$ \mathbb{I}(\partial_u, \partial_u) = \nabla_{\partial_u} \partial_u = \frac{1}{2} \mathbf{n}, $$

where $\nabla$ is the Levi-Civita connection of $\hat{\mathbf{H}}^{2,n}$.
*Proof.* For the last assertion,

$$ egin{cases} 
\nabla_{\partial_u} \partial_u = \frac{1}{2}(\exp(u)z_1 - \exp(v)z_2 + \exp(-u)z_3 - \exp(-v)z_4), \\
\nabla_{\partial_v} \partial_u = \nabla_{\partial_u} \partial_v = 0, \\
\nabla_{\partial_v} \partial_v = -\nabla_{\partial_u} \partial_u.
\end{cases} $$

□

**4.1.2. Geodesics.** Barbot surfaces are isometric to the Euclidean plane. Their intrinsic geometry is simple, and we investigate now their extrinsic geometry by the asymptotic behavior of geodesics in the boundary of $H^{2,n}$.

**Proposition 4.1.3 (Behavior of geodesics at infinity of a Barbot surface).** There are four ideal points in $S(\infty)$ such that any representing ray of these ideal points converges in $S \cup C$ to a unique point of the interior of an edge. Every other ray converges to a vertex (see Figure 1).

<figure><img src="image_6.png" /><figcaption>FIGURE 1. Illustration of the behavior of geodesics in a Barbot surface with initial velocity between two singular directions (in red) at two different points.</figcaption></figure>

We call the four directions going to edges singular.

*Proof.* We first do the proof at the point $x = z_1 + z_2 + z_3 + z_4$. Let $X = (u, v)$. The geodesic ray with initial velocity $X$ at $x$ is given by the equation (2).

Suppose $u = \pm v$. Then $\lim_{t \to +\infty} \exp(-|u|t)\sigma(t)$ equals

* $z_1 + z_2$ if $u = v > 0$,

* $z_2 + z_3$ if $u = -v$ and $u < 0$,

* $z_3 + z_4$ if $u = v < 0$,

* $z_4 + z_1$ if $u = -v$ and $u > 0$.

Suppose now that $u \neq \pm v$. Then $\lim_{t \to +\infty} \exp(-\max(|u|, |v|)t)\sigma(t)$ equals

* $z_1$ if $u > |v|$,

* $z_2$ if $v > |u|$,

* $z_3$ if $-u > |v|$,

* $z_4$ if $-v > |u|$.

Hence the four singular directions, at $x$, are directed by the vectors $(1, 1)$, $(-1, 1)$, $(-1, -1)$, $(1, -1)$.

Now if $y$ is another point, $y = a \cdot x$ for an element $a$ of $A_0$. Since $a$ acts by isometries on $S$, stabilizes the vertices, sends points of an edge to points on the same edge transitively, this finishes the proof. □
At a point $x$, there are four singular directions, orthogonal to each other. The four directions at $x$ making an angle $\pi/4$ with the singular directions are called *regular*. They are four ideal points and we have the following

**Corollary 4.1.4.** *The angle between two consecutive regular ideal points in a Barbot surface is $\pi/2$.*

**4.1.3.** *Space-horofunctions in Barbot surfaces.*

**Proposition 4.1.5.** *After a parametrization*

$$f: \mathbb{R}^2 \to \hat{\mathbf{H}}^{2,n} \\ (u, v) \mapsto \exp(u)z_1 + \exp(v)z_2 + \exp(-u)z_3 + \exp(-v)z_4,$$

of a Barbot surface, the space-horofunction associated to a vertex, say $v_1$, has a representative $h$ defined by

$$h \circ f((u, v)) = -u,$$

and the space-horofunction associated to a point on an edge, say $P(z_1 + z_2)$, has a representative $\tilde{h}$ defined by

$$\tilde{h} \circ f((u, v)) = \log(\exp(-t) + \exp(-s)).$$

**Remark 4.1.6.** In the case of vertices, the space-horofunction equals the Busemann function associated to the regular direction tending to the vertex. The horoballs associated to are half planes, and their boundary join the two neighboring vertices. See Figure 2.

<figure><img src="image_7.png" /><figcaption>FIGURE 2. Space-horoball (in gray) in a Barbot surface, associated to a vertex, with the regular direction $\xi$.</figcaption></figure>

The space-horoballs associated with vertices in a maximal surface have the same geometric shape as in the Barbot case, as we shall see.

**4.2. Renormalization.** The aim of this Section is to show that polygonal surfaces are asymptotically flat. This is done using

* a projection between maximal surfaces,

* the renormalization process.

**4.2.1. The radial projection.** Here we define the radial projection between two maximal surfaces. Take $\Sigma_1$ and $\Sigma_2$ two maximal surfaces of $\mathbf{H}^{2,n}$. We will project $\Sigma_1$ to $\Sigma_2$ following a direction in $N_x\Sigma_1$ at each point $x$ of $\Sigma_1$.

**Definition 4.2.1.** The radial projection from $\Sigma_1$ to $\Sigma_2$ is the map that sends a point $x$ in $\Sigma_1$ to the unique point of $\Sigma_2$ in the timelike sphere orthogonal to $T_x\Sigma_1$.

The map is well defined by [LTW22, Proposition 3.10 (iii)].

We will always denote by $\rho$ the radial projection between two maximal surfaces. The timelike spheres orthogonal to $T_x\Sigma_1$ vary smoothly with the point $\sigma$, so the projection $\rho$ is smooth.
### 4.2.2. Asymptotics of a maximal surface near a vertex.
Let $\Sigma$ be a maximal surface with space boundary $\Lambda$. Let $v$ be a vertex of $\Lambda$ (recall Definition 3.1.2). By Corollary 3.2.6, there is a point $v_4$ in $\text{Ein}^{1,n}$ such that $v_1, v_2, v_3, v_4$ are the vertices of a Barbot crown $C$ coinciding with $\Lambda$ between $v_1, v_2$ and $v_3$, where $v_2 = v$ and $v_1, v_3$ are the points at the end of the two edges passing through $v$. We fix some notations:

* $S$ is the Barbot surface with space boundary $C$,

* $\gamma$ is the connected component of $\Lambda \setminus \{v_1, v_3\}$ containing $v_2$ (see Figure 3),

* $\bar{\Sigma}$ and $\bar{S}$ are the $C^0$ surfaces with boundary $\Sigma \cup \gamma$ and $S \cup \gamma$, respectively,

* $\rho: S \to \Sigma$ denotes the radial projection (see Definition 4.2.1),

* $\Lambda$ is the Cartan subgroup of $G$ determined by the crown $C$ (see Section 3.2.2).

<figure><img src="image_8.png" /><figcaption>FIGURE 3. A semi-positive loop $\Lambda$ with a Barbot crown coinciding with near a vertex.</figcaption></figure>

We now prove the following

**Proposition 4.2.2.** The radial projection $\rho: S \to \Sigma$ extends as a continuous map $\bar{\rho}: \bar{S} \to \bar{\Sigma}$ that restricts to the identity on $\gamma$.

*Proof.* Let $(x_k)_{k \in \mathbb{N}}$ be a sequence of points of $S$ tending to a point $p$ in $\gamma$. We fix a hyperbolic basis $(z_1, z_2, z_3, z_4)$ associated to the vertices $v_i$, and $x = [z_1+z_2+z_3+z_4]$ in $S$. We have, for each integer $k$, real numbers $l_k, m_k$ such that

$$x_k = a(l_k, m_k) \cdot x = [l_k z_1 + m_k z_2 + l_k^{-1} z_3 + m_k^{-1} z_4].$$

Since $(x_k)_{k \in \mathbb{N}}$ tends to $p$ in $\gamma$, we have

(1) the sequence $(m_k)_{k \in \mathbb{N}}$ tends to $+\infty$,

(2) the sequence $(l_k m_k)_{k \in \mathbb{N}}$ is bounded by below,

(3) the sequence $(l_k^{-1} m_k)_{k \in \mathbb{N}}$ is bounded by below.

Now we look at the sequence of points $(y_k)_{k \in \mathbb{N}}$ in $\Sigma$ defined by

$$y_k := \rho(x_k) \quad \forall k \in \mathbb{N}.$$

We will prove that the sequence $(y_k)_{k \in \mathbb{N}}$ converges to $p$.

**First step.** We claim that the timelike distance between $x_k$ and $y_k$ tends to 0 as $k$ goes to $+\infty$.

We look at the sequence $a_k = a(l_k^{-1}, m_k^{-1})$ of elements of $\Lambda$. By definition, for all integers $k$, $a_k \cdot x_k = x$ and by the very definition of the radial projection, $a_k \cdot y_k$ is a point in the timelike sphere $\exp_x(\mathbb{N}_x S)$. By the renormalization lemma near a vertex (Lemma 3.3.1), up to extracting a subsequence, $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ converges to
C as $k$ goes to $+\infty$. Hence, by Theorem 2.2.7, the sequence of maximal surfaces $(a_k \cdot \Sigma)_{k \in \mathbb{N}}$ subconverges to $S$. Moreover, since $a_k \cdot y_k$ is in the timelike sphere defined by $x$ and $S$ for each integer $k$, we have

$$ \lim_{k \to +\infty} a_k \cdot y_k = x. $$

Since $a_k$ is an isometry, the timelike distance between $x_k$ and $y_k$ equals the timelike distance between $x = a_k \cdot x_k$ and $a_k \cdot y_k$, hence tends to 0.

This concludes the first step.

To complete the proof of the proposition, we take coherent lifts $\hat{S}, \hat{\Sigma}, (\hat{x}_k)_{k \in \mathbb{N}}$ and $(\hat{y}_k)_{k \in \mathbb{N}}$ of $S, \Sigma, (x_k)_{k \in \mathbb{N}}$ and $(y_k)_{k \in \mathbb{N}}$ in $\hat{H}^{2,n}$, respectively.

**Second step.** We claim that the sequence $(\hat{y}_k)_{k \in \mathbb{N}}$, up to the multiplication by positive constants, tends to a vector in $p$.

For each integer $k$, we write $\hat{x}_k = \exp(l_k)z_1 + \exp(m_k)z_2 + \exp(-l_k)z_3 + \exp(-m_k)z_4$. The geodesic segment between $\hat{x}_k$ and $\hat{y}_k$ is parametrized by

$$ [0, d_k] \to \hat{H}^{2,n} \\ t \mapsto \cos(t)\hat{x}_k + \sin(t)n_k, $$

with $n_k$ a unit vector tangent to $\hat{H}^{2,n}$ at the point $\hat{x}_k$ and orthogonal to $\mathcal{T}_{\hat{x}_k} \hat{S}$. We know by the previous lemma that the number $d_k$ which equals the timelike distance between $\hat{x}_k$ and $\hat{y}_k$ tends to 0 as $k$ goes to $+\infty$.

The surface $\hat{S}$ lies inside a totally geodesic copy of $\hat{H}^{2,1}$, that we call abusively $\hat{H}^{2,1}$. This copy defines a splitting $E = (F \perp G)$ where $H^{2,1}$ lies in $\mathbf{P}(F)$ and $\mathbf{q}_G$ is of non-degenerate signature $(0, n-1)$. Let us decompose orthogonality $n_k$ as

$$ n_k = n_k^{\perp} + n_k^{\perp}, $$

where $n_k^{\perp}$ is the part of $n_k$ tangent to $\hat{H}^{2,1}$ and $n_k^{\perp}$ the part of $n_k$ normal to $\hat{H}^{2,1}$, so $n_k^{\perp}$ lies in $G \simeq \mathbb{R}^{0,n-1}$. Since $-1 = \mathbf{q}(n_k) = \mathbf{q}(n_k^{\perp}) + \mathbf{q}(n_k^{\perp})$, we have

$$ -1 \le \mathbf{q}(n_k^{\perp}) \le 0 \quad \text{and} \quad -1 \le \mathbf{q}(n_k^{\perp}) \le 0. $$

By continuity, and since $n_k^{\perp}$ lies in the unit $\mathbf{q}$-ball of $G$,

$$ \lim_{k \to +\infty} \sin(d_k)n_k^{\perp} = 0. $$

If $n_k^{\perp} \ne 0$ for an infinite number of indices $k$ in $\mathbb{N}$, we extract a subsequence where $n_k^{\perp} \ne 0$. By Proposition 4.1.2,

$$ \frac{n_k^{\perp}}{\mathbf{q}(n_k^{\perp})} = \pm (\exp(l_k)z_1 - \exp(m_k)z_2 + \exp(-l_k)z_3 - \exp(-m_k)z_4). $$

Since $(x_k)_{k \in \mathbb{N}}$ tends to the point $p$ in $\gamma$, there exists positive real numbers $c_k$ for each integer $k$, and $z = az_1 + bz_2 + cz_3$ a vector in $p\setminus\{0\}$, such that

$$ \lim_{k \to +\infty} c_k \hat{x}_k = z. $$

Then, we have the limit

$$ \lim_{k \to +\infty} c_k \frac{n_k^{\perp}}{\mathbf{q}(n_k^{\perp})} = \pm (az_1 - bz_2 + cz_3). $$

In each case, by continuity,

$$ c_k \hat{y}_k = c_k \cos(d_k) \hat{x}_k + c_k \sin(d_k) n_k^{\perp} + c_k \sin(d_k) n_k^{\perp} \underset{k \to +\infty}{\longrightarrow} z $$

because the first term tends to $z$ and the other ones to zero.

This concludes the second step, and the proof of the proposition. $\square$
**4.2.3. Renormalization.** We take here the same notations as in the previous paragraph.

**Lemma 4.2.3 (Renormalization).** Let $(y_k)_{k \in \mathbb{N}}$ be a sequence of elements of $\Sigma$ tending to a point $p$ in $\gamma$ and $x$ be a point of $S$. Then there is a sequence $(a_k)_{k \in \mathbb{N}}$ of elements of $\mathcal{A}$ such that $((a_k \cdot \Sigma, a_k \cdot y_k))_{k \in \mathbb{N}}$ subconverges to the pointed Barbot surface $(S, x)$.

By Proposition 4.2.2, the radial projection $\rho : S \to \Sigma$ extends as a continuous map from $\bar{S} = S \cup \gamma$ to $\bar{\Sigma} = \Sigma \cup \gamma$. The map $\rho$ realizes a homeomorphism from a neighborhood $\mathcal{U}$ of $\gamma$ in $\bar{S}$ to a neighborhood $\mathcal{V}$ of $\gamma$ in $\bar{\Sigma}$. Denote by $\bar{\rho}$ the obtained homeomorphism from $\mathcal{U}$ to $\mathcal{V}$.

*Proof.* Fix the point $x$ in $S$. This defines a hyperbolic basis $(z_1, z_2, z_3, z_4)$ such that $z_i$ are vectors in $v_i$ and $x = z_1 + z_2 + z_3 + z_4$. Since $(y_k)_{k \in \mathbb{N}}$ converges to $p$ in $\gamma$, there is an integer $K$ such that the point $y_k$ lies in $\mathcal{V}$ for all $k \ge K$. Hence we can define for all $k \ge K$

$$x_k := \bar{\rho}^{-1}(y_k).$$

The sequence $(x_k)_{k \in \mathbb{N}}$ converges to $p$. Hence for all $k \ge K$,

$$x_k = a(l_k, m_k) \cdot x,$$

with $a(l_k, m_k)$ elements of the Cartan subgroup $\mathcal{A}$ written in the hyperbolic basis $(z_1, z_2, z_3, z_4)$, satisfying the following conditions

(1) the sequence $(m_k)_{k \in \mathbb{N}}$ tends to $+\infty$,

(2) the sequence $(l_k m_k)_{k \in \mathbb{N}}$ is bounded by below,

(3) the sequence $(l_k^{-1} m_k)_{k \in \mathbb{N}}$ is bounded by below.

Denote by $(a_k)_{k \in \mathbb{N}}$ the sequence of elements of $\mathcal{A}$ defined by $a_k := a(l_k^{-1}, m_k^{-1})$ for $k \ge K$. By the renormalization lemma (Lemma 3.3.1), the sequence of polygons $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ subconverges to $C$. Moreover, since for all $k \ge K$, $a_k \cdot x_k = x$ and $y_k$ is the radial projection of $x_k$, $a_k \cdot y_k$ is in the timelike sphere $\exp(\mathbb{N}_x S)$. By Theorem 2.2.7, the sequence of maximal surfaces $(a_k \cdot \Sigma)_{k \in \mathbb{N}}$ subconverges to the maximal surface with space boundary $C$, that is $S$. The limit point $\lim a_k \cdot y_k$ is $x$ since $\exp(\mathbb{N}_x S)$ is closed and intersects $S$ in the unique point $x$. $\square$

**Corollary 4.2.4.** Polygonal surfaces are asymptotically flat.

*Proof.* Let $\Sigma$ be polygonal. The renormalization lemma shows that for every diverging sequence $(y_k)_{k \in \mathbb{N}}$ in $\Sigma$, the sequence $((\Sigma, y_k))_{k \in \mathbb{N}}$ subconverges, up to renormalization, to a pointed Barbot surface $(S, x)$. Hence, $\lim_{k \to +\infty} K_{\Sigma}(y_k) = K_S(x) = 0$. $\square$

**4.3. Space-horofunctions.** The results of this Section are the followings. Let $\Sigma$ be a maximal surface of $\mathbb{H}^{2,n}$ with space boundary $\Lambda$. Suppose that $\Sigma$ is not a Barbot surface. For each point $p$ in $\Lambda$ the space-horofunction associated with $p$ satisfies the following two properties:

* the function $h_p$ is strictly quasi-convex,

* the horospheres $\{h_p = C\}$ are curves that tend, in $\partial^s \Sigma$, to the two points at the end of an edge with $p$ (see Figure 4).

**4.3.1. Quasiconvexity.** Let $\Sigma$ be a Hadamard surface.

**Definition 4.3.1.** A function $h$ on $\Sigma$ is said to be (strictly) quasiconvex if its sublevel sets are (strictly) convex subsets of the surface.

It is equivalent, for a $\mathcal{C}^2$-function, to each of the following two properties:
* for every $x, y$ points of $\Sigma$, denoting $\gamma: [0, 1] \to \Sigma$ the geodesic segment joining $x$ and $y$, we have $h(\gamma(t)) \le \max(h(x), h(y))$ for $0 < t < 1$ (put a strict inequality for strict quasiconvexity),

* for every critical direction $X$, meaning $dh \cdot X = 0$, the Hessian of $h$ satisfies $\nabla^2 h \cdot (X, X) \ge 0$ (put a strict inequality for strict quasiconvexity).

**Theorem 4.3.2.** Let $\Sigma$ be a maximal surface of $\mathbf{H}^{2,n}$ that is not a Barbot surface. Take a point $p$ in the space boundary $\Lambda$ of $\Sigma$. Then the space-horofunction $(h_p)|_{\Sigma}$ is strictly quasiconvex.

For the proof, we denote by $h$ the restriction of $h_p$ to $\Sigma$. We now compute the gradient and Hessian of the space-horofunction $h$ with respect to the induced metric on $\Sigma$.

**Proposition 4.3.3** (Gradient and Hessian of space-horofunctions). The gradient and the Hessian of the space-horofunction $h$ are, at $x$ and for vectors $X, Y$ in $\mathcal{T}_x\Sigma$:

* $\mathrm{grad}(h)_x = z^\top/\langle x, z \rangle$,

* $\nabla_x^2 h \cdot (X, Y) = \langle X, Y \rangle_x + \frac{\langle \mathbb{I}(X, Y)_x, z \rangle}{\langle x, z \rangle} - (X \cdot h)_x (Y \cdot h)_x$,

where $z$ is a non zero vector in $p$, and $z^\top$ is the part of $z$ tangent to $\mathcal{T}_x\Sigma$ when we decompose $\mathbf{E} = x \oplus \mathcal{T}_x\Sigma \oplus \mathcal{N}_x\Sigma$.

*Proof.* We fix a lift $\hat{\Sigma}$ of $\Sigma$ in $\hat{\mathbf{H}}^{2,n}$ to make computations and $z$ in $p$. There is a representative of $h$ that takes the value

$$ \log(-\langle x, z \rangle), $$

at any point $x$ in $\hat{\Sigma}$. We still call this function $h$. Hence, if $X$ is a tangent vector of $\hat{\Sigma}$ at $x$,

$$ d_x h \cdot X = \frac{\langle X, z \rangle}{\langle x, z \rangle}, $$

and the gradient of $h$ equals

$$ \mathrm{grad}(h)_x = \frac{z^\top}{\langle x, z \rangle}. $$

Denote by $D$ the flat connection on $\mathbf{E}$. It splits, at a point $x$, as

$$ D_X Y = \langle X, Y \rangle_x + \nabla_X Y + \mathbb{I}(X, Y), $$

in the splitting $\mathbf{E} = x \oplus \mathcal{T}_x\Sigma \oplus \mathcal{N}_x\Sigma$, where $\nabla$ is the Levi-Civita connection of $\hat{\Sigma}$ and $\mathbb{I}$ the second fundamental form of $\hat{\Sigma}$ in $\hat{\mathbf{H}}^{2,n}$.

Now, to compute the Hessian, recall that for any $X, Y$ vector fields on $\hat{\Sigma}$,

$$ \nabla^2 h \cdot (X, Y) = X \cdot (Y \cdot h) - \nabla_X Y \cdot h. $$

We first compute $X \cdot (Y \cdot h)$ for two vector fields $X, Y$ on $\hat{\Sigma}$. It gives

$$ \begin{aligned} X \cdot (Y \cdot h)(x) &= X \cdot \frac{\langle Y, z \rangle}{\langle x, z \rangle} \\ &= \frac{\langle x, z \rangle \langle D_X Y, Z \rangle - \langle Y, z \rangle \langle X, z \rangle}{\langle x, z \rangle^2} \\ &= \frac{\langle\langle X, Y \rangle_x x + (\nabla_X Y)_x + \mathbb{I}_x(X, Y), z \rangle}{\langle x, z \rangle} - \frac{\langle Y, z \rangle_x \langle X, z \rangle_x}{\langle x, z \rangle^2}. \end{aligned} $$

Hence the result

$$ \nabla_x^2 h \cdot (X, Y) = \langle X, Y \rangle_x + \frac{\langle \mathbb{I}_x(X, Y), z \rangle}{\langle x, z \rangle} - (X \cdot h)_x (Y \cdot h)_x. $$

□
*Proof of Theorem 4.3.2.* Labourie and Toulisse have showed ([LT23, Lemma 5.12]) that the function

$$ \beta : (x, X) \mapsto \frac{\langle \mathbb{I}_x(X, X), z \rangle}{\langle x, z \rangle}, $$

defined on the unit tangent bundle of a maximal surface Σ, satisfies

$$ \beta(x, \cdot)^2 \le (\mathbf{q}(\mathrm{grad}(h))_x - 1)(1 + K_{\Sigma}(x)). $$

By [LT23, Theorem 5.5], the norm of the gradient of a space-horofunction satisfies the inequalities:

$$ (3) \qquad 0 < \mathbf{q}(\mathrm{grad}(h)) \le 2. $$

Since $\mathbf{q}(\mathrm{grad}(h)) \le 2$ and $K_{\Sigma} \le 0$, we have

$$ \beta \ge -1. $$

For a maximal surface that is not a Barbot surface, the strict inequality $K_{\Sigma} < 0$ gives

$$ \beta > -1. $$

Putting these inequalities in the formula of the Hessian of $h$ we obtain that if $X$ is a critical direction at a point $x$, on a Barbot surface

$$ \nabla_x^2 h \cdot (X, X) \ge 0, $$

making $h$ quasiconvex, and if $\Sigma$ is not a Barbot surface,

$$ \nabla_x^2 h \cdot (X, X) > 0, $$

making $h$ strictly quasiconvex.

<figure><img src="image_9.png" /></figure>

**4.3.2. The shape of space-horoballs.** Let $\Sigma$ be a maximal surface, denote by $\Lambda$ its space boundary. We study in this paragraph the shape of space-horoballs and space-horospheres associated to a point $p$ in $\Lambda$.

**Definition 4.3.4.** *Space-horoballs and space-horospheres* are sublevel sets and level sets of a space-horofunction, respectively.

The geometry of space-horoballs is well understood on hyperbolic planes where they are exactly the intrinsic horoballs, and on Barbot surfaces (see Paragraph 4.1.3). First we have the following proposition.

**Proposition 4.3.5.** *The points at infinity of a space-horosphere $\{h_p = C\}$ are in an edge with $p$.*

*Proof.* Denote $\gamma = \{h_p = C\}$ the level set for the value $C$ in $\mathbb{R}$. The closure $\bar{\gamma}$ of $\gamma$ in $\bar{\Sigma} = \Sigma \cup \Lambda$ meets $\Lambda$. We want to show that the points in the frontier $\bar{\gamma} \setminus \gamma$ are in an edge with $p$. Let $(x_k)_{k \in \mathbb{N}}$ be a sequence of points of $\gamma$ tending to a point $q$ at infinity. Take a lift $(\hat{x}_k)_{k \in \mathbb{N}}$ in $\hat{\mathbf{H}}^{2,n}$. There exists a sequence of non-zero real numbers $(\lambda_k)_{k \in \mathbb{N}}$, tending to 0, such that $(\lambda_k \hat{x}_k)_{k \in \mathbb{N}}$ tends to $w$ a vector in $q \setminus \{0\}$ as $k \to +\infty$. Denote by $z$ a non-zero vector in $p$ such that $h(x) = \log(|\langle \hat{x}, z \rangle|)$ is a representative of $h_p$. Now we have

$$ |\langle \lambda_k \hat{x}_k, z \rangle| = |\lambda_k| |\langle \hat{x}_k, z \rangle| = |\lambda_k| \exp(c). $$

Taking the limit $k \to +\infty$ we obtain $\langle w, z \rangle = 0$, so $q$ is in an edge with $p$. $\square$

<figure><img src="image_10.png" /></figure>

The proposition has an important corollary.

**Definition 4.3.6.** A point $p$ in a semi-positive loop $\Lambda$ is said to be a *positive point* if it does not belong to any edge in $\Lambda$.
**Corollary 4.3.7.** Let $\Sigma$ be a maximal surface of $\mathbf{H}^{2,n}$, with space boundary $\Lambda$ in $\mathbf{Ein}^{1,n}$. If $p$ is a positive point in $\Lambda$, then the space-horoballs associated to $p$ are strictly convex subsets of $\Sigma$. The frontier of a space-horoball in $\Sigma \cup \Lambda$ is a circle containing only $p$ in $\Lambda$.

We will need more work to understand the shape of space-horoballs associated to vertices. Recall that we call a vertex, in a semi-positive loop $\Lambda$, a point that meets two different segment of photons in $\Lambda$. If $v$ is a vertex, it is in the extremity of two different photons and we denote by $v_-$ and $v_+$ the two opposite extremities.

**Proposition 4.3.8.** Let $\Sigma$ be a maximal surface of $\mathbf{H}^{2,n}$, with space boundary $\Lambda$ in $\mathbf{Ein}^{1,n}$. If $v$ is a vertex of $\Lambda$, then the space-horoballs associated to $v$ are strictly convex subsets of $\Sigma$ and their boundaries, space-horospheres associated to $v$, have boundary $\{v_+, v_-\}$ at infinity.

See Figure 4.

<figure><img src="image_11.png" /><figcaption>FIGURE 4. Shape of space-horoballs in a maximal surface. On the left the situation where $p$ is a positive point, on the right the situation where $p$ is a vertex.</figcaption></figure>

*Proof.* Fix $z$ in $v$ so a representative $h$ of $h_v$ takes the value

$$h(y) = \log(-\langle \hat{y}, z \rangle)$$

at points $y$ of $\Sigma$, where $\hat{y}$ is a lift of $y$ to the double cover $\hat{\mathbf{H}}^{2,n}$ satisfying $\langle \hat{y}, z \rangle < 0$. Denote by $H = \{h = C\}$ a space-horosphere. Take a sequence $(y_k)_{k \in \mathbb{N}}$ of points in $H$ tending to a point $q$ in an edge $\phi$ with $v$. We want to show that $q$ is one of the two points $v_+, v_-$.

Suppose that $q$ is in the interior of $\phi$. By Corollary 3.2.6 and Proposition 4.2.2, there is a Barbot surface $S$ such that the radial projection $\rho: S \to \Sigma$ is a diffeomorphism in a neighborhood of $[v, q]$ in $\phi$ and extends as the identity on $[v, q]$. Denote by $\bar{\rho}$ the homeomorphism extending $\rho$ in a neighborhood of $[v, q]$. Then there is an integer $K$ such that $y_k$ is in the range of $\bar{\rho}$ for all $k \ge K$. Define, for all $k \ge K$, the point $x_k := \bar{\rho}^{-1}(y_k)$ in $S$. We take coherent lifts to the double cover to make computations, and denote with the same symbols the lifted surfaces and points. We have

$$y_k = \cos(\theta_k)x_k + \sin(\theta_k)n_k,$$

where $n_k$ is a unit vector tangent to $\hat{\mathbf{H}}^{2,n}$ at the point $x_k$ and normal to $T_{x_k}S$, and $\theta_k$ tends to 0 when $k$ tends to $+\infty$. Hence

$$h(x_k) = \log \left( \left\langle \frac{(y_k - \sin(\theta_k)n_k)}{\cos(\theta_k)}, z \right\rangle \right).$$
So $h(x_k)$ tends to $C$ when $k \to +\infty$ by continuity. But this is a contradiction because $(x_k)_{k \in \mathbb{N}}$ tending to $q$ in the interior of $\phi$ implies that $(h(x_k))_{k \in \mathbb{N}}$ tends to $-\infty$ (see the shape of space-horospheres in a Barbot surface in paragraph 4.1.3).

At the end, $q$ must be $v_+$ or $v_-$.

<figure><img src="image_12.png" /></figure>

**Corollary 4.3.9.** Let $\Sigma$ be a maximal surface of $H^{2,n}$ with space boundary $\Lambda$ in $\text{Ein}^{1,n}$. Suppose that $v_1, v_2, v_3$ are three consecutive vertices of $\Lambda$. Then the space-horoballs associated to $v_1$ and $v_3$ are either disjoint or their intersection is a non bounded strictly convex set containing only $v_2$ at infinity. Moreover, if we parametrize by $\gamma$ the space-horosphere associated to $v_1$, the distance between $\gamma(t)$ and any space-horosphere associated to $v_3$ is bounded by a constant depending only on the chosen space-horospheres.

*Proof.* Fix $z_1$ in $v_1$ and $z_3$ in $v_3$ such that $\langle z_1, z_3 \rangle = -1/4$. This choice fixes representatives $h_1$ and $h_3$ of $h_{v_1}$ and $h_{v_3}$ respectively so that the exterior space-horospheres are defined by $H_i = \{h_i = d_i\}$. By Corollary 3.2.6 and Proposition 4.2.2, there is a Barbot surface $S$ such that the radial projection $\rho: S \to \Sigma$ is a diffeomorphism in a neighborhood of $v_2$. Moreover $\rho$ extends as the identity on this neighborhood. Denote by $\bar{\rho}$ the obtained homeomorphism from a neighborhood of $v_2$ in $\bar{\Sigma}$ to a neighborhood of $v_2$ in $\bar{S}$.

Take an increasing sequence $(t_k)_{k \in \mathbb{N}}$ of real numbers tending to $+\infty$. Then $\gamma(t_k)$ tends to $v_2$. Denote $x_k := \bar{\rho}^{-1}(\gamma(t_k))$ for each $k$ big enough. Since $\gamma(t_k)$ is the radial projection of $x_k$, the value $h_i(x_k)$ tends to $d$ when $k$ tends to $+\infty$. We renormalize using elements of the Cartan subgroup $\Lambda$ associated to the Barbot surface $S$. We write these elements in an hyperbolic basis $(z_1, z_2, z_3, z_4)$ and define $x = z_1 + z_2 + z_3 + z_4$. We have $x_k = a(l_k, m_k) \cdot x$ for uniquely determined positive real numbers $l_k, m_k$ and $a(l_k, m_k)$ defined as in Section 3.2.2. Define $a_k = (1, m_k^{-1})$. Each isometry $a_k$ preserves $z_i$ and $z_{i+2}$, hence sends $H_i$ to the corresponding level set $\{h_i = d_i\}$ in $a_k \cdot \Sigma$ for $i = 1, 3$. At the limit, by renormalization (Lemma 4.2.3), $a_k \cdot \Sigma$ tends to $S$ and $a_k \cdot H_i$ tends to $\{h_i = d_i\}$ (for $i = 1, 3$) in the Barbot surface. The two level sets $\{h_1 = d_1\}$ and $\{h_3 = d_3\}$ have a fixed distance (they are parallel geodesics), hence the result.

□

<figure><img src="image_13.png" /></figure>

**4.4. The ideal boundary of asymptotically flat maximal surfaces.** Let $\Sigma$ be an asymptotically flat maximal surface of $H^{2,n}$ with space boundary $\Lambda$. We study the $\text{Tits}$ distance of the ideal boundary $\Sigma(\infty)$ of $\Sigma$, using the previous sections. See Appendix A for details about the ideal boundary and the $\text{Tits}$ distance.

4.4.1. *Polygonal case.* Let $v_-, v, v_+$ be three adjacent vertices of $\Lambda$. We denote by $h_-, h, h_+$ representatives of space-horofunctions associated to $v_-, v, v_+$.

**Definition 4.4.1.** A geodesic ray $c$ of $\Sigma$ is said to be *regular associated to* $v$ if $t \mapsto d(c(t), B)$ is bounded for every space-horoball $B$ associated to $v_-$ or $v_+$.

**Theorem 4.4.2.** Let $\Sigma$ be a maximal surface with space boundary $\Lambda$ and $v$ be a vertex of $\Lambda$. For each point $x$ in $\Sigma$, there is a unique regular geodesic ray associated to $v$ starting from $x$. Moreover, two regular geodesic rays associated to $v$ are asymptotic.

The collection of regular rays associated to a vertex $v$ defines, by the previous theorem, a point in $\Sigma(\infty)$, called the *regular ideal point associated to* $v$.

*Proof.* By Proposition 4.3.8 and Corollary 4.3.9, the space-horoballs associated to the vertices $v_+$ and $v_-$ are strictly convex subsets of $\Sigma$ and their intersection is either empty or is a non bounded convex set containing only $v$ at infinity with boundary two curves at finite distance.
We now prove the existence. Take $x$ in $\Sigma$. Denote by $\gamma_-$ and $\gamma_+$ the exterior space-horospheres passing through $x$ and associated to $v_-$ and $v_+$, respectively. These curves bound a convex non bounded region denoted $R$. We restrict $\gamma_+$ and $\gamma_-$ to the part between $x$ and $v$. Denote by $\mathbf{S}^1$ the unit circle of $(\Gamma x\Sigma, q_{|\Gamma x\Sigma})$, and parametrize it by arclength $\theta$ in $[0, 2\pi]$. Denote by $\theta_-$ the direction of $\gamma_-$ at $x$ and $\theta_+$ the direction of $\gamma_+$ at $x$ such that $\theta_- \le \theta_+$ and every direction $\theta$ in $(\theta_-, \theta_+)$ points inside $R$. The angles $\theta_-$ and $\theta_+$ are different because $\gamma_-$ and $\gamma_+$ are not tangent at $x$. The set of geodesics rays starting from $x$, pointing inside $R$ and crossing $\gamma_-$ corresponds to an interval $(\theta_-, t_-)$ (it is defined by an open condition). The set of geodesic rays starting from $x$, pointing inside $R$ and crossing $\gamma_+$ corresponds to an interval $(t_+, \theta_+)$, see figure 5.

We claim that $t_- \le t_+$.

Indeed, if $\theta_- < \theta < t_-$, the corresponding ray $c_\theta$ crosses $\gamma^-$, so it goes out of the sublevel set $\{h_- \le h_-(x)\}$ and never goes inside after, by convexity. So $c_\theta$ cannot cross $\gamma^+$, and $\theta < t^+$. Hence $t^- \le t^+$.

Every geodesic ray emanating from $x$ with a direction in $[t^-, t^+]$ stays inside $R$, so converges to $v$ at infinity, by staying at finite distance of $\gamma^-$ and $\gamma^+$, hence is regular.

Now, since $\gamma^-$ and $\gamma^+$ are at bounded distance, $t^- = t^+$. This proves existence and uniqueness.

To prove that the collection of regular rays defines an ideal point, first remark that if we take a geodesic ray $c$ with initial direction $\dot{c}(0)$ a regular direction, then the velocity of the geodesic ray $\dot{c}(t)$ is a regular direction for every $t > 0$, by definition. Finally, two regular geodesic rays tending to the same vertex $v$ stay at finite distance, by definition, so they are asymptotic and the collection of regular geodesic rays tending to $v$ is a point of the ideal boundary $\Sigma(\infty)$. $\square$

<figure><img src="image_14.png" /><figcaption>FIGURE 5. Construction of a regular geodesic ray emanating from $x$.</figcaption></figure>

**Remark 4.4.3.** The above construction is $G$ invariant because $G$ sends space-horoballs to space-horoballs hence regular geodesic rays to regular geodesic rays.

Suppose now that $v_-, v_2, v_3, v_+$ are four adjacent vertices of $\Lambda$. Denote by $\xi_2$ and $\xi_3$ the regular points of $\Sigma(\infty)$ associated to $v_2$ and $v_3$, respectively.

**Theorem 4.4.4** (Tits distance between neighboring vertices). *The Tits distance between $\xi_2$ and $\xi_3$ equals $\pi/2$.*

*Proof.* Take a regular geodesic ray $c$ associated to $v_2$. By the properties of the angle distance (see Lemma A.1.7), we have the formula

$$ \angle(\xi_2, \xi_3) = \lim_{t \to +\infty} \angle_{c(t)}(\xi_2, \xi_3). $$

For any increasing sequence $(t_k)_{k \in \mathbb{N}}$ of positive real numbers tending to $+\infty$, we can renormalize along $(c(t_k))_{k \in \mathbb{N}}$ that converges to $v_2$. To this aim, use a sequence
$(a_k)_{k \in \mathbb{N}}$ that stabilizes $v_-$, $v_2$, $v_3$, and such that $(a_k \cdot \Lambda)_{k \in \mathbb{N}}$ tends to a Barbot crown $C$ (hence $(a_k \cdot v_+)_{k \in \mathbb{N}}$ tends to the fourth vertex of $C$) and $(a_k \cdot c(t_k))_{k \in \mathbb{N}}$ converges to a point $x$ of the Barbot surface $S$ associated to $C$. At each step, the regular ray $c_k = c(\cdot + t_k)$ starting from $c(t_k)$ and associated to $v_2$ is sent to the regular geodesic ray starting at $a_k \cdot c(t_k)$ and associated to $v_2$. Also at each step, the regular ray $\gamma_k$ starting from $c(t_k)$ and associated to $v_3$ is sent to the regular geodesic ray starting at $a_k \cdot c(t_k)$ and associated to $v_3$.

At the limit, the sequence $(c_k)_{k \in \mathbb{N}}$ converges to the regular geodesic ray $c_\infty$ starting at $x$ associated to $v_2$ and the sequence $(\gamma_k)_{k \in \mathbb{N}}$ converges to the regular geodesic ray $\gamma_\infty$ starting at $x$ associated to $v_3$. By Corollary 4.1.4, $\not\wedge_x(c_\infty(0), \gamma_\infty(0)) = \pi/2$, hence

$$ \not\wedge(\xi_2, \xi_3) = \pi/2. $$

By Lemma A.1.8, and since the angle distance is smaller than $\pi$, the Tits distance between $\xi_2$ and $\xi_3$ is

$$ \mathrm{Td}(\xi_2, \xi_3) = \not\wedge(\xi_2, \xi_3) = \pi/2. $$

□

**Positive case.** For this paragraph, suppose that $\Lambda$ contains an open positive non-empty interval $I$. By positive, we mean that two distinct points of $I$ span a plane of signature $(1, 1)$ and it is equivalent to the fact that $I$ does not contain any edge.

We can now assign to every point $p$ in $I$ a collection of ideal points in a natural way, similar to the construction of regular geodesic rays associated to a vertex.

**Definition 4.4.5.** A geodesic ray $c$ of $\Sigma$ is said to be *regular* with respect to $p$ if it stays inside the space-horoball $\{h_p \le h_p(c(0))\}$.

**Theorem 4.4.6.** For each point $p$ of $I$, there is an ideal point $\xi$ consisting of regular rays associated to $p$.

*Proof.* By Corollary 4.3.7, the space-horoballs associated to $p$ are strictly convex non-bounded subsets of $\Sigma$ containing only $p$ at infinity.

Fix a point $x_0$ of $\Sigma$, and look at the set of geodesic rays starting from $x_0$, pointing inside the space-horoball and crossing the space-horosphere $\{h_p = h_p(x_0)\}$. The collection of such rays is the union of two intervals in the unit tangent space of $\Sigma$ at $x_0$. Between these intervals is the direction of at least one ray $c_0$ staying inside the space-horoball. Denote by $\xi$ the associated ideal point. Because of the shape of the space-horoball, the ray converges uniquely to $p$ at infinity.

Now we prove that every ray with direction $\xi$ is regular, that is, stays inside space-horoballs associated to $p$. If $c$ is a geodesic ray starting at $x$ in $\Sigma$ with direction $\xi$, then it stays at finite distance from the initial ray $c_0$. By strict convexity of the space-horoballs, if $c$ goes out of the space-horoball $\{h_p \le h_p(x)\}$, the distance between $c(t)$ and $\{h_p \le h_p(x)\}$ goes to $+\infty$ when $t$ goes to $+\infty$. Since two space-horoballs are at finite distance (the norm of $\mathrm{grad}(h_p)$ is bounded by above), if $c$ goes out of the space-horoball $\{h_p \le h_p(x)\}$, the distance between $c(t)$ and $\{h_p \le h_p(x_0)\}$ also diverges. But $c_0$ is included in $\{h_p \le h_p(x_0)\}$, hence the result. □

**Remark 4.4.7.** This theorem is weaker than the one in the vertex case. They may be several different rays staying inside a space-horoball at a point. It will nevertheless be sufficient for our purpose.

**Corollary 4.4.8.** Geodesic rays have a unique limit point at infinity in the positive sector $I$.

*Proof.* Indeed, for each point $p$ of $I$ there is a regular geodesic ray tending uniquely to $p$. Every geodesic ray pointing through the positive sector will be stuck between regular rays. □
By the previous theorem, we associate to each point $p$ in $I$ a set of ideal points: the regular rays tending to $p$. We will now study the *Tits distance* between these ideal points.

**Theorem 4.4.9 (Tits distance between positive points).** Let $p, q$ be two distinct points in $I$. The *Tits distance* between ideal points associated to $p$ and $q$ is infinite.

*Proof.* We will first show that the angle distance between every two regular ideal points associated to points $p$ and $q$ in $I$ is $\pi$. Let $p$ and $q$ be distinct points in $I$, and denote by $\xi_p, \xi_q$ some regular ideal points associated to $p, q$, respectively. Remark that the property of being regular is invariant by the action of $G$. Indeed an isometry $g$ in $G$ sends geodesics to geodesics and space-horofunctions to space-horofunctions. Suppose that $\nparallel(\xi_p, \xi_q) < \pi$, and that $p$ and $q$ are close enough so that for every point $r$ in $I$ between $p$ and $q$, and any regular ideal point $\xi_r$ associated to $r$, we have

$$ \nparallel(\xi_p, \xi_q) = \nparallel(\xi_p, \xi_r) + \nparallel(\xi_r, \xi_q). $$

Indeed, by Lemma A.1.10, there is a unique *Tits geodesic segment* between $\xi_p$ and $\xi_q$ in $\Sigma(\infty)$ and $\xi_r$ is inside this segment.

Take now a point $v_4$ in the connected component of $\Lambda\setminus\{p, q\}$ that does not contain $r$. The lines $r$ and $v_4$ span a plane of signature $(1, 1)$ in $\mathbb{E}$. Indeed, by Proposition 2.2.3, if $r$ and $v_4$ span a photon, then $\Lambda$ must contain an edge containing $r$ and $v_4$. Then there are $v_1$ and $v_3$ two points of $\mathbb{Ein}^{1,n}$ such that $v_1, r, v_3, v_4$ are the vertices of a Barbot crown $C$. Denote by $A$ the Cartan subgroup associated to $C$. In the associated Barbot surface $S$, we look at a regular geodesic $\gamma$ tending to $r$. Hence $\gamma$ can be parametrized by $\gamma(s) = a(t, s)$, where $t$ is a real constant. We use the radial projection $\rho$, Definition 4.2.1, between $S$ and $\Sigma$, to project $\gamma$ to a curve $\rho(\gamma)$ on $\Sigma$.

We claim that $\lim_{s \to +\infty} \rho(\gamma)(s) = r$. Indeed, take any sequence $(s_k)_{k \in \mathbb{N}}$ of real numbers tending to $+\infty$. For each integer $k$, the geodesic of $\mathbb{H}^{2,n}$ between $\gamma(s_k)$ and $\rho(\gamma)(s_k)$ is timelike, making a sequence of timelike geodesics indexed by $k$. Recall that a timelike geodesic in $\mathbb{H}^{2,n}$ is the projectivization of a plane of signature $(0, 2)$. Up to a subsequence, this sequence of geodesics converges to a plane that cannot be of signature $(1, 1)$. Since $(\gamma(s_k))_{k \in \mathbb{N}}$ tends to $r$, the only possibility is that $(\rho(\gamma)(s_k))_{k \in \mathbb{N}}$ converges to $r$.

We want to show that $\nparallel(\xi_p, \xi_q)$ is indeed $\pi$ (contradicting our assumption that it is less than $\pi$). To this aim, we show that the angle $\nparallel_{\rho(\gamma)(s)}(\xi_p, \xi_q)$ is as close as we want from $\pi$.

By supposition, the angle $\nparallel(\xi_p, \xi_q)$ is less than $\pi$, hence at each point $\rho(\gamma)(s)$ we have

$$ \nparallel_{\rho(\gamma)(s)}(\xi_p, \xi_q) = \nparallel_{\rho(\gamma)(s)}(\xi_p, \xi_r) + \nparallel_{\rho(\gamma)(s)}(\xi_r, \xi_q) < \pi. $$

We now renormalize by the action of $(a(t, s_k))_{k \in \mathbb{N}}$, where $(s_k)_{k \in \mathbb{N}}$ is any sequence of real numbers tending to $+\infty$. Since $\Sigma$ is asymptotically flat, the sequence $(a(t, s_k) \cdot \Sigma, a(t, s_k) \cdot \rho(\gamma)(s_k))$ converges to a pointed Barbot surface $(S_\infty, x_\infty)$. By proximality, $a(t, s_k) \cdot p$ and $a(t, s_k) \cdot q$ tend to $v_4$ when $k \to +\infty$. Since $G$ preserves geodesics and space-horofunctions, the limit of geodesic rays with direction $\xi_p$, at $\rho(\gamma)(s_k)$, when renormalized, points inside the space-horoball associated to $v_4$ at $x_\infty$. The same occurs for $\xi_q$. The geodesic rays with direction $\xi_r$ converge to a geodesic ray pointing inside the space-horoball, at $x_\infty$, associated to $r$. Because of the shape of space-horoballs, the angle between $\xi_p$ and $\xi_q$ must be $\pi$, contradicting the assumption (see Figure 6).

At the end,

$$ \nparallel(\xi_p, \xi_q) = \pi. $$

This is true for every pair of ideal points $(p, q)$ close enough in $I^2$ and every chosen regular ray $\xi_p, \xi_q$ tending to $p, q$, hence the result. $\square$
<figure><img src="image_15.png" /><figcaption>FIGURE 6. Renormalization in a positive interval and angle between two regular ideal points</figcaption></figure>

**4.5. Polygonal surfaces and finite total curvature.** Let $\Sigma$ be a maximal surface of $H^{2,n}$ and denote by $\Lambda$ its space boundary. We prove in this section the following theorem.

**Theorem 4.5.1.** The *Tits distance on $\Sigma(\infty)$ has finite perimeter, hence finite total curvature, if and only if $\Sigma$ is a polygonal surface. A polygonal surface with $N+4$ vertices has total curvature equal to $-\frac{\pi}{4}N$.*

Suppose that $\Lambda$ is a lightlike polygon with $N+4$ vertices. As shown in Theorem 4.4.2 and Theorem 4.4.4, they are $N+4$ distinct ideal points $\xi_1, \dots, \xi_{N+4}$, corresponding to vertices, such that $\mathrm{Td}(\xi_i, \xi_{i+1}) = \pi/2$ for each $i$. Then the *Tits distance* $(\Sigma(\infty), \mathrm{Td})$ is that of a circle of perimeter $(N+4)\pi/2$. Using the formula in Theorem A.3.3 we obtain that

$$ \int_{\Sigma} K = 2\pi - (N+4)\frac{\pi}{2} = -\frac{\pi}{2}N. $$

Suppose now that $\Lambda$ is not a lightlike polygon. If $\Lambda$ contains an infinite number of adjacent segment of photons, the arguments used in the polygonal case show that the total curvature of $\Sigma$ is $-\infty$. Suppose that $\Lambda$ does not contain an infinite number of adjacent photons. Then $\Lambda$ contains a positive open non-empty interval, denoted $I$.

**Lemma 4.5.2.** If $K$ does not tend to $0$ at infinity between two rays $c_1, c_2$ at a point $x$ in $\Sigma$, then the sector determined by $c_1, c_2$ and $x$ has infinite total curvature and the *Tits distance* between $c_1(\infty)$ and $c_2(\infty)$ is infinite.

*Proof.* Let $(x_k)_{k \in \mathbb{N}}$ be a diverging sequence of $\Sigma$ such that $(K_{x_k})_{k \in \mathbb{N}}$ tends to a negative number. We claim that there exist $\varepsilon > 0$ and $-c < 0$ such that $K < -c$ on each ball centered at $x_k$ with radius $\varepsilon$.

If not, we can find a sequence $(y_k)_{k \in \mathbb{N}}$ of points of $\Sigma$ such that $(d(x_k, y_k))_{k \in \mathbb{N}}$ tends to 0 and $(K_{y_k})_{k \in \mathbb{N}}$ tends to 0. Hence the sequence of pointed maximal surfaces $((\Sigma, y_k))_{k \in \mathbb{N}}$ subconverges uniformly on compact sets to a pointed Barbot crown. Then we obtain that a subsequence of $(K_{x_k})_{k \in \mathbb{N}}$ tends to 0, contradiction. Hence the total curvature is infinite. $\square$

So we can suppose that $\Sigma$ is asymptotically flat.

By Theorem 4.4.9, for each point of $I$ there is an associated ideal point $\xi$ such that the *Tits distance* between each pair of these points is infinite. Hence, by Theorem A.3.3 we have

$$ \int_{\Sigma} K = -\infty, $$

and this finishes the proof of Theorem 4.5.1.
## 5. THE QUARTIC DIFFERENTIAL

This section is devoted to the study of the quartic differential associated with a maximal surface of $H^{2,n}$. We study the link between the induced metric and the quartic metric. By showing that they are close to each other, we infer that their ideal boundaries are isometric.

### 5.0.1. Uniformization.
As a first consequence of Theorem 4.5.1,

**Corollary 5.0.1.** A polygonal surface is of parabolic type and asymptotically flat.

*Proof.* By [BF41], a simply connected non-compact Riemannian surface with finite total curvature must be of parabolic type.

If there is a diverging sequence of points $(x_k)_{k \in \mathbb{N}}$ of a maximal surface $\Sigma$ such that the curvature $K$ along this sequence tends to a negative constant $C$, then $\Sigma$ is not polygonal by the proof of Lemma 4.5.2 and Theorem 4.5.1. $\square$

### 5.1. The quartic differential associated to maximal surfaces.
We denote now by $g$ the induced metric on the tangent space of a spacelike surface $\Sigma$ and by $g_N$ the induced metric on its normal bundle $N\Sigma$. If $\Sigma$ is a maximal spacelike surface of $H^{2,n}$, there is a holomorphic object on $(\Sigma, [g])$ one can build using the second fundamental form.

#### 5.1.1. Definition of the quartic differential.
The second fundamental form $\mathbb{I}$ of $\Sigma$ is a symmetric 2 tensor with values in the normal bundle $N\Sigma$. By complexifying, we obtain a 2-tensor

$$ \mathbb{I}^C : \mathbb{T}^C \Sigma \times \mathbb{T}^C \Sigma \to \mathbb{N}^C \Sigma, $$

that we can decompose under types using the complex structure coming from $g$ ($T^{0,1}\Sigma = \mathbb{T}^{0,1}\Sigma \oplus \mathbb{T}^{1,0}\Sigma$). The (1,1) part of $\mathbb{I}^C$ is null because it equals the $g$-trace of $\mathbb{I}$ and $\Sigma$ is maximal. The bundle

$$ E = (\mathbb{T}^{1,0}\Sigma)^{\vee} \otimes (\mathbb{T}^{1,0}\Sigma)^{\vee} \otimes \mathbb{N}^C \Sigma $$

is endowed with the connection $\nabla^E$ coming from the complexification of the Levi-Civita connection of $\Sigma$ on the tangent bundle and from the normal connection of $\Sigma$ on the normal bundle. The (0,1) part $\tilde{\partial}^E$ of the associated exterior derivative $d_{\nabla^E}$ is a Dolbeault operator. The (2,0) part of $\mathbb{I}^C$ is $\tilde{\partial}^E$ holomorphic (see [LTW22, Lemma 5.6] for a proof).

**Definition 5.1.1.** The holomorphic quartic differential associated to $(\Sigma, [g])$ is defined by

$$ \varphi_4 := g_N^C (2\mathbb{I}^{2,0}, 2\mathbb{I}^{2,0}) \in \mathbb{H}^0(\mathcal{K}^4). $$

Now we give an expression for $\varphi_4$. First, denote by $J$ the complex structure associated with the induced metric $g$. Let $X, Y$ be complex vector fields on $\Sigma$. They can be decomposed in (1,0) and (0,1) type and we have

$$ \begin{aligned} 2\mathbb{I}^{2,0}(X, Y) &= 2\mathbb{I}^C(X^{1,0}, Y^{1,0}) \\ &= \frac{1}{2}\mathbb{I}^C(X - iJX, Y - iJY) \\ &= \mathbb{I}(X, Y) - i\mathbb{I}(JX, Y), \end{aligned} $$

where for the last equality we used the fact that $\mathbb{I}(J\cdot, J\cdot) = -\mathbb{I}(\cdot, \cdot)$.

Secondly, we compute $\varphi_4$ in a holomorphic chart. Take $z = x + iy$ a holomorphic coordinate centered at a point $p$ of $\Sigma$. The metric $g$ reads in this chart

$$ g_z = \exp(2f(z))(dx^2 + dy^2), $$

with $f$ a smooth function on $\mathbb{C}$. We now have

$$ 2\mathbb{I}^{2,0} = 2\mathbb{I}^{2,0}(\tilde{\partial}_z, \tilde{\partial}_z)dz^2, $$
and

$$2\mathbb{I}^{2,0}(\partial_z, \partial_z) = \mathbb{I}(\partial_x, \partial_x) - i\mathbb{I}(\partial_x, \partial_y).$$

Hence, taking the complex extension of $g_N$, we obtain in the chart $z$

$$\varphi_4 = \varphi_4(z)dz^4,$$

with

$$(\varphi) \qquad \varphi_4(z) = g_N(\mathbb{I}(\partial_x, \partial_x), \mathbb{I}(\partial_x, \partial_x)) - g_N(\mathbb{I}(\partial_x, \partial_y), \mathbb{I}(\partial_x, \partial_y)) \\ \qquad \qquad - 2ig_N(\mathbb{I}(\partial_x, \partial_x), \mathbb{I}(\partial_x, \partial_y)).$$

Denote by $f$ the homomorphism $\mathbb{I}(\partial_x, \cdot) : (\mathbb{T}\Sigma, g) \to (\mathbb{N}\Sigma, -g_N)$. The last equation says that at a point $z$ where $\varphi_4(z) = 0$ the map $f$ is conformal:

$$\begin{cases} g_N(f(\partial_x), f(\partial_x)) = g_N(f(\partial_y), f(\partial_y)), \\ g_N(f(\partial_x), f(\partial_y)) = 0. \end{cases}$$

5.1.2. **The metric associated to a quartic differential.** We associate a singular metric to $(\Sigma, \varphi_4)$. In our setting, we define

$$g_4 := |\varphi_4|^{1/2}$$

and call $g_4$ the *quartic metric*.

This defines a smooth flat metric on $\Sigma$ outside the zeroes of $\varphi_4$, with conical singularity of angle $(k+4)\frac{\pi}{2}$ at each zero of order $k$. See Appendix B for precise definitions.

**5.2. Conformal type and quartic differential of an asymptotically flat maximal surface.** We know that polygonal surfaces are of parabolic type. The aim of this section is to prove that asymptotically flat maximal surfaces are parabolic and have polynomial quartic differential.

**Proposition 5.2.1.** Let $\Sigma$ be asymptotically flat. For every $\varepsilon > 0$ there is a compact set $K$ in $\Sigma$ such that $g$ and $g_4$ are $(1+\varepsilon)$-biLipschitz outside $K$.

*Proof.* In a Barbot surface, the quartic differential equals $dz^4$, as one can see using Equation $(\varphi)$ and the computations in Proposition 4.1.2. Hence, in this case, the quartic metric equals the flat induced metric.

Now, for every diverging sequence of points $(x_k)_{k \in \mathbb{N}}$ in $\Sigma$, we can take a sub-sequence such that $((\Sigma, x_k))_{k \in \mathbb{N}}$ subconverges, up to renormalization, to a pointed Barbot surface. $\square$

We now follow the arguments of [DW15, Section 7] to prove that an asymptotically flat maximal surface is of parabolic type and has polynomial quartic differential. We put the proof with our notations.

**Lemma 5.2.2 (Lemma 7.3 in [DW15]).** The quartic differential $\varphi_4$ has a finite number of zeros, and the quartic metric $g_4$ is complete.

*Proof.* The previous lemma shows that the zero set of $\varphi_4$ lies in a compact set. It is discrete because $\varphi_4 \neq 0$. Outside a compact $K$, $g$ and $g_4$ are bi-Lipschitz. If $c$ is a $g_4$-geodesic, it is a $g$-quasi geodesic outside $K$, hence is complete. $\square$

**Lemma 5.2.3 (Lemma 7.5 in [DW15]).** The Riemann surface $(\Sigma, [g])$ is of parabolic type.

*Proof.* If not, then $(\Sigma, [g])$ is bi-holomorphic to the unit disc. We write $\varphi_4$ in the $z$-coordinate as

$$\varphi_4 = P(z) \exp(f(z)) dz^4,$$

with $P$ a polynomial and $f$ holomorphic since $\varphi_4$ has a finite number of zeroes.
The metric $|\exp(f)|^{1/2}|dz|^2$ cannot be complete because the unit disc is not bi-holomorphic to the complex plane. Hence there is a divergent path $\gamma$ of finite $|\exp(f)|^{1/2}|dz|^2$-length.

Now the $g_4$-length of $\gamma$ is also finite because $P$ is bounded on the unit disc. Hence $g_4$ is not complete, contradicting the previous lemma. □

**Theorem 5.2.4 (Lemma 9.6 in [Oss86]).** The quartic differential $\varphi_4$ is polynomial.

*Proof.* We identify $(\Sigma, [g])$ with the complex plane with $z$-coordinate. So $\varphi_4 = \varphi_4(z)dz^4$. Since $\varphi_4$ has a finite number of zeros,

$$ \varphi_4 = P(z) \exp(f(z)) dz^4, $$

with $f$ a holomorphic function on $\mathbb{C}$. We will prove that $f$ is constant. If $P$ is of degree $N$ and $M$ is an integer bigger than $N/4$, then

$$ |\varphi_4(z)|^{1/4} = O(|z^M \exp(f(z))|). $$

Since $g_4$ is complete, for every divergent path $\gamma$, the $g_4$-length $\int_{\gamma} |\varphi_4(z)|^{1/4} |dz|$ diverges, hence also $\int_{\gamma} |z^M \exp(f(z))| |dz|$. Now take

$$ h(z) := \int_{[0,z]} \zeta^M \exp(f(\zeta)) d\zeta. $$

The function $h$ has a zero of order $M+1$ in $z=0$, hence

$$ h(z) = z^{M+1}(b_0 + b_1 z + \dots), $$

with $b_0 = a_0/(M+1) \neq 0$. Define another parameter $w$ by a single valued branch of $h^{1/(M+1)}$:

$$ w(z) := h^{1/(M+1)}(z) = z(b_0 + b_1 z + \dots)^{1/(M+1)}. $$

The parameter $w$ is well defined in a neighborhood of 0, and since $w'(0) = b_0^{1/(M+1)} \neq 0$ it has an inverse $z = \psi(w)$ where $\psi$ is defined in a neighborhood of zero. We show now that $\psi$ is defined globally. Take a path $\gamma(t) = tw_0$ where $w_0$ is in $\mathbb{C}^*$ and $t$ in $[0, 1)$. The composition $\psi \circ \gamma$ is a path in the $z$-plane. Since $d(w^{M+1})(z) = dh(z) = z^M \exp(f(z))$, we have

$$ \int_{\gamma} |z^M \exp(f(z))| |dz| = \int_{\gamma} |d(w^{M+1})| = |w_0|^{M+1}. $$

Since $\psi \circ \gamma$ has finite $|z^M \exp(f(z))| |dz|$-length, hence finite $g_4$-length, it is not divergent in the $z$-plane. So $\psi \circ \gamma(t)$ has some adherence value when $t \to 1$ and since $(\psi \circ \gamma)'(1) = \psi'(w_0) \neq 0$, the path $\psi \circ \gamma$ can be continued in $t=1$. Hence, $\psi$ is an automorphism of $\mathbb{C}$, so $w$ as a function of $z$. The function $w(z) = h^{1/(M+1)}(z)$ being linear implies that $h$ is a polynomial in $z$. Hence $h'(z) = z^M \exp(f(z))$ is polynomial and $f$ is constant. □

**5.3. Comparison between the space boundary and the quartic boundary.**
Let $\Sigma$ be an asymptotically flat maximal surface. By Theorem 5.2.4, denoting by $g$ the induced metric on $\Sigma$, the Riemann surface $(\Sigma, [g])$ is biholomorphic to the complex plane and has polynomial quartic differential $\varphi_4$. As a consequence, by Appendix B, the metric $g_4$ associated with $\varphi_4$ has an ideal boundary of perimeter $(N+4)\pi/2$ for the Tits metric.

In this section, using a careful comparison between the induced metric $g$ and the quartic metric $g_4$, we prove the following.

**Theorem 5.3.1.** An asymptotically flat maximal surface of $\mathbf{H}^{2,n}$ with polynomial quartic differential of degree $N$ is a polygonal surface with $N+4$ vertices.
To this aim, we compare the Tits distances on the ideal boundaries of $(\Sigma, g)$ and of $(\Sigma, g_4)$. Denote by $d$ and $d_4$ the distance functions associated with $g$ and $g_4$, respectively.

**Lemma 5.3.2.** For every positive number $\varepsilon$, there exists a nonnegative number $b$ such that the identity map is a $(1 + \varepsilon, b)$ quasi isometry between $(\Sigma, d)$ and $(\Sigma, d_4)$.

*Proof.* Fix a positive number $\varepsilon$. We know, by Proposition 5.2.1 that $g$ and $g_4$ are $(1 + \varepsilon)$-biLipschitz outside a compact set $K$ in $\Sigma$.

We define

$$b := \sup_{(x,y) \in K^2} \{|d(x, y) - d_4(x, y)|\}.$$

Take $x, y$ in $\Sigma$, and denote by $c$ the $g$-geodesic between $x$ and $y$ of length $L = d(x, y)$. If $c([0, L]) \cap K$ is empty, then the $g_4$ length of $c$ is smaller than $(1 + \varepsilon)L$, hence

$$d_4(x, y) \le (1 + \varepsilon)d(x, y).$$

If $c([0, L]) \cap K$ is not empty, then denote by $t_1$ the smallest real number such that $c(t_1) \in K$ and by $t_2$ the biggest real number such that $c(t_2) \in K$. Then by triangular inequality

$$d_4(x, y) \le d_4(x, c(t_1)) + d_4(c(t_1), c(t_2)) + d_4(c(t_2), y).$$

Now, by definition of $b$, and since $c([0, t_1))$ and $c((t_2, L])$ do not intersect $K$, we have

$$d_4(x, y) \le (1 + \varepsilon)t_1 + (t_2 - t_1) + b + (1 + \varepsilon)(L - t_2) \le (1 + \varepsilon)L + b.$$

Using the same arguments, we obtain

$$d(x, y) \le (1 + \varepsilon)d_4(x, y) + b,$$

so $d$ and $d_4$ are $(1 + \varepsilon, b)$-quasi isometric. $\square$

<figure><img src="image_16.png" /></figure>

We can now use the results of Section A.2 to prove that $(\Sigma, g)$ and $(\Sigma, g_4)$ have isometric Tits boundaries.

**Lemma 5.3.3.** The *Tits boundary of* $(\Sigma, g)$ *is isometric to a circle of perimeter* $\frac{\pi}{2}(N + 4)$.

*Proof.* By Proposition B.1.3, $(\Sigma, g_4)$ has a *Tits boundary isometric to a circle of perimeter $(N + 4)\frac{\pi}{2}$*. Applying the previous Lemma and Corollary A.2.2 gives the result. $\square$

<figure><img src="image_17.png" /></figure>

*Proof of Theorem 5.3.1.* By Theorem 4.5.1 and the previous lemma, $\Sigma$ is a polyg- onal surface with $N + 4$ vertices. $\square$

<figure><img src="image_18.png" /></figure>

## 6. MAIN THEOREM

The Main Theorem is a consequence of Theorem 4.5.1, Theorem 5.2.4 and Theorem 5.3.1. We recall:

**Theorem 6.0.1 (Main Theorem).** Let $\Sigma$ be a maximal surface in $H^{2,n}$, and denote by $g$ its induced metric. The following are equivalent:

(i) The space boundary of $\Sigma$ is a lightlike polygon with finitely many vertices,

(ii) The total curvature of $(\Sigma, g)$ is finite,

(iii) The surface $(\Sigma, g)$ is asymptotically flat.

Moreover, if the conditions are satisfied, and denoting by $N + 4$ the number of vertices of $\partial^s\Sigma$, the Riemann surface $(\Sigma, [g])$ is of parabolic type, has polynomial quartic differential of degree $N$ and the total curvature of $\Sigma$ equals $-\frac{\pi}{2}N$.
*Proof.* Let $\Sigma$ be a maximal surface in $\mathbb{H}^{2,n}$. By Theorem 4.5.1, the two first items are equivalent, and if $\Sigma$ is polygonal with $N + 4$ vertices then its total curvature equals $-\frac{\pi}{2}N$. By Corollary 4.2.4 a polygonal surface is asymptotically flat. By Theorem 5.2.4, an asymptotically flat maximal surface has parabolic type and polynomial quartic differential. At the end, if $\Sigma$ is of parabolic type and has polynomial quartic differential of degree $N$, then by Theorem 5.3.1 it is polygonal with $N + 4$ vertices. $\square$

## APPENDIX A. THE GEOMETRY OF HADAMARD MANIFOLDS

In this section, we investigate the link between the Tits distance $T_d$, defined in the ideal boundary of a Hadamard surface $\Sigma$, and the global geometry of the surface. We are interested in two results:

* the metric space $(\Sigma(\infty), T_d)$ can be approximated by spheres of growing radius, renormalized by the radius, via the $l$-distance,

* the Tits distance has finite perimeter $P$ if and only if the surface has finite total curvature $2\pi - P$.

**A.1. The Tits distance.** For this section, let $M$ be a Hadamard manifold.

**A.1.1. The angle distance.**

**Definition A.1.1.** Let $\xi, \eta$ be two ideal boundary points of $M$. The angle distance between $\xi$ and $\eta$, denoted $\angle(\xi, \eta)$, is defined by

$$ \angle(\xi, \eta) := \sup_{x \in M} \angle_x(\xi, \eta), $$

where the angle $\angle_x(\xi, \eta)$ is defined to be the usual Riemannian angle between the two rays $\xi_x, \eta_x$ corresponding to $\xi, \eta$, at $x$, respectively.

The angle distance is indeed a distance in $M(\infty)$.

**Remark A.1.2.** The angle distance on $M(\infty)$ equals the usual round distance on the unit sphere if and only if $M$ is the euclidean space. In the hyperbolic space, the angle distance is $\pi$ between any two different points of the ideal boundary: indeed two ideal points can always be joined by a geodesic in this case. In general, the angle distance helps to detect flatness or hyperbolicity of Hadamard manifolds, as we shall see.

**A.1.2. The $l$-distance.** Another interesting distance can be defined on $M(\infty)$. The idea is to look at the renormalized distance function restricted to spheres of growing radius. Take two rays $c_1, c_2$ starting from the same point in $M$. Using the triangular inequality, we obtain that for every $0 < s < t$ we have

$$ 0 \le \frac{d(c_1(s), c_2(s))}{s} \le \frac{d(c_1(t), c_2(t))}{t} \le 2. $$

**Definition A.1.3.** Let $\xi_1, \xi_2$ be two ideal boundary points of $M$, and $c_1, c_2$ be two rays starting at the same point representing $\xi_1$ and $\xi_2$, respectively. The $l$-distance between $\xi_1$ and $\xi_2$, denoted $l(\xi_1, \xi_2)$, is defined by

$$ l(\xi_1, \xi_2) = \lim_{t \to +\infty} \frac{d(c_1(t), c_2(t))}{t}. $$

This definition does not depend on the chosen point, the function $l$ is a distance and is related to the angle distance by the following formula ([BGS85, Lemma 4.4]).

**Proposition A.1.4.** If $\xi$ and $\eta$ are two ideal boundary points of $M$, then

$$ l(\xi, \eta) = 2 \sin \left( \frac{\angle(\xi, \eta)}{2} \right). $$
This formula is the same that links the restriction of the Euclidean distance on a sphere of radius 1 to the spherical distance.

A.1.3. **Inner distances.** We make a brief aside to explain the construction of the inner distance associated to a metric space ([BGS85, 4.6]). Let (A, d) be a metric space. If $c: [0, 1] \to A$ is a continuous curve, we can define the length of c as being the supremum over all subdivisions $0 = t_0 \le t_1 \le \dots \le t_k = 1$ of the sums

$$ \sum_{i=1}^{k} d(c(t_{i-1}), c(t_i)). $$

If $x, y$ are in A, we define $d_i(x, y)$ the inner distance between $x$ and $y$ by

$$ d_i(x, y) = \inf\{L(c) \mid c \text{ is a continuous curve joining } x \text{ and } y\}. $$

All existing curve between $x$ and $y$ may be of infinite length, in which case $d_i(x, y) = \infty$. A distance $d$ is said to be inner if the corresponding inner distance $d_i$ equals $d$.

**Remark A.1.5.** The distance of a Riemannian manifold is an inner distance. Also in general the inner distance $d_i$ of any distance $d$ is an inner distance $((d_i)_i = d_i)$.

### A.1.4. The *Tits distance*.

**Definition A.1.6.** The *Tits distance* on $M(\infty)$, denoted Td, is defined to be the inner distance of the angle distance,

$$ Td = \mathcal{A}_i. $$

It turns out that the *Tits distance* is also the inner distance associated with $l$ ([BGS85, Lemma 4.8]). Hence they are two ways to understand the *Tits distance* on the ideal boundary of a Hadamard manifold, compute the angle distance or compute the $l$-distance. In this article, we use both the angle distance and the $l$-distance, each being more appropriate in some context.

A.1.5. *Computations.* We will need several properties of the *Tits distance* to make computations. The first is [BGS85, Lemma 4.2].

**Lemma A.1.7.** Let $\xi, \eta$ be two ideal boundary points of M. Then we can compute the angle distance between $\xi$ and $\eta$ as follows:

$$ \mathcal{A}(\xi, \eta) = \lim_{t \to +\infty} \mathcal{A}_{c(t)}(\xi, \eta), $$

where $c: [0, +\infty) \to \mathcal{A}(\xi, \eta)$ is a geodesic ray of direction $\xi$.

Then [BGS85, Lemma 4.7].

**Lemma A.1.8.** If $\xi, \eta$ are two ideal boundary points of M such that there is no geodesic $c$ of M satisfying $c(+\infty) = \xi$ and $c(-\infty) = \eta$, then $Td(\xi, \eta) = \mathcal{A}(\xi, \eta)$. In particular, if $\mathcal{A}(\xi, \eta) < \pi$, then $\mathcal{A}(\xi, \eta) = Td(\xi, \eta)$.

As a consequence, we have [BGS85, Lemma 4.10]:

**Lemma A.1.9.** If $Td(\xi, \eta) > \pi$, then there is a geodesic $c$ in M with $c(+\infty) = \xi$ and $c(-\infty) = \eta$.

Since $(M(\infty), Td)$ is a metric space, we can ask for the existence and uniqueness of geodesics between two points. We have [BGS85, Lemma 4.11].

**Lemma A.1.10.** Any two points in $M(\infty)$ with finite *Tits distance* can be joined by a minimal geodesic of $M(\infty)$. Moreover, the geodesic is unique if the *Tits distance* is smaller than $\pi$.
### A.2. Compact Tits boundaries and nets.
We study in this Section the case of
Tits boundary of Hadamard surfaces, and more precisely when the Tits distance is
compact.

**Theorem A.2.1.** *Suppose that $M$ is of dimension 2. Then the following conditions are equivalent.*

(i) The metric space $(M(\infty), \mathrm{Td})$ is compact,

(ii) The metric space $(M(\infty), \mathrm{Td})$ is isometric to a circle of radius $R \ge 1$,

(iii) The metric space $(M(\infty), l)$ admits a finite 1-net.

Moreover, in the case $(M(\infty), \mathrm{Td})$ is isometric to a circle of radius $R$, we have for every point $o$ in $M$

$$2\pi R = \lim_{r \to +\infty} \frac{P_o(r)}{r},$$

where $P_o(r)$ denotes the perimeter of the circle of radius $r$ in $(M, d)$ centered at $o$.

A $\varepsilon$-net in a metric space $(E, d)$, following [Gro07, Definition 2.14], is a subset $N$ of $E$ such that $d(x, N) < \varepsilon$ for every $x$ in $E$. We can replace 1-net by $\delta$-net with any $\delta < 2$ in this Theorem, but the value 1 is more convenient.

The last theorem does not hold in higher dimensions. For example, the Tits boundary of a higher rank symmetric space of noncompact type always admits a finite $\sqrt{2}$-net, but is not compact. Indeed, length spaces of dimension 1 are determined by their length and nothing else, but as soon as the dimension is bigger than 1, the situation is much more complex.

*Proof.* (i) $\Rightarrow$ (ii). Suppose that $(M(\infty), \mathrm{Td})$ is compact. Then the natural map $\varphi : (\mathbb{T}_x^1 \Sigma, \mathbb{A}_x) \to (\Sigma(\infty), \mathrm{Td})$ is a homeomorphism and the Tits boundary is homeomorphic to a circle. Now $\mathrm{Td} : \Sigma(\infty)^2 \to \mathbb{R}$ is continuous, hence bounded, and $(M(\infty), \mathrm{Td})$ has finite diameter $D \ge \pi$. As a consequence, since $\mathrm{Td}$ is an inner distance, $(M(\infty), \mathrm{Td})$ has finite perimeter $P = 2D$ and is isometric to a circle of radius $P/2\pi \ge 1$.

(ii) $\Rightarrow$ (iii) is immediate.

In order to prove (iii) $\Rightarrow$ (i), we recall the following fact. By Proposition A.1.4 and Lemma A.1.8, if the $l$ distance between two points $\xi, \eta$ of $M(\infty)$ is strictly smaller than 2, then

$$\mathrm{Td}(\xi, \eta) = 2 \arcsin \left( \frac{l(\xi, \eta)}{2} \right) < \pi$$

and by Lemma A.1.10 the Tits geodesic between $\xi$ and $\eta$ in $M(\infty)$ is unique.

Suppose that $(M(\infty), l)$ admits a finite 1-net $N = \{\xi_1, \dots, \xi_K\}$. Then $N$ is a $\frac{\pi}{3}$-net in $(M(\infty), \mathrm{Td})$. This implies that $(M(\infty), \mathrm{Td})$ is compact. Indeed take a sequence of points $(\eta_k)_{k \in \mathbb{N}}$ in $M(\infty)$. There is an integer $i$ between 1 and $K$ such that an infinite number of $\eta_k$ is at Tits distance less than $\frac{\pi}{3}$. This provides us a subsequence $(\eta_{\psi(k)})_{k \in \mathbb{N}}$ included in $\{\eta \in M(\infty) \mid \mathrm{Td}(\eta, \xi_i) \le \frac{\pi}{3}\}$ and this last set is homeomorphic to a segment of length $\frac{2\pi}{3}$, yielding the result.

In order to prove the last assertion, it suffices to use the following facts:

* If $(M(\infty), \mathrm{Td})$ is compact, the sequence of metric spaces $(S_o(r), d/r)$ converges, in the Gromov-Hausdorff sense, to $(M(\infty), l)$ when $r$ goes to $+\infty$, here $o$ is a point of $M$ and $S_o(r)$ is the circle centered at $o$ and of radius $r$,

* a sequence of length spaces converges to a length space ([Gro07, Proposition 3.8]).

Hence the length space obtained from $(S_o(r), d/r)$, that is a circle of perimeter $P_o(r)/r$, converges to $(M(\infty), \mathrm{Td})$. $\square$
**Corollary A.2.2.** Let $\Sigma$ be diffeomorphic to $\mathbb{R}^2$, and $d_1, d_2$ be two Hadamard distances in $\Sigma$. Suppose that for every positive number $\varepsilon$ the identity map is a $(1 + \varepsilon, b)$-quasi isometry between $(\Sigma, d_1)$ and $(\Sigma, d_2)$, for a nonnegative number $b$. Then $(\Sigma(\infty), Td_1)$ is compact if and only if $(\Sigma(\infty), Td_2)$ is compact. Moreover, if $(\Sigma(\infty), Td_1)$ and $(\Sigma(\infty), Td_2)$ are compact, they are isometric.

We fix some notations. Let us fix $o$ a point in $\Sigma$. We denote by $S_i(r)$ the circle of radius $r$ centered at $o$ in $(\Sigma, d_i)$. We define a map to compare the circles $S_1(r)$ and $S_2(r)$ as follows.

If $x$ is a point of $S_1(r)$, denote by $c_{ox}$ the $d_2$-geodesic of $\Sigma$ from $o$ to $x$. This geodesic intersects the sphere $S_2(r)$ at a unique point $y$, and we define $f_r(x) = y$. More precisely,

$$f_r : S_1(r) \to S_2(r) \\ x \mapsto c_{ox} \left( \frac{r}{d_2(o, x)} \right).$$

For topological reasons, the map $f_r$ is surjective.

**Lemma A.2.3.** For every positive number $\varepsilon$, there is a positive number $r_0$ and a function $\delta : [r_0, +\infty) \to \mathbb{R}_+$ satisfying $\lim_{r \to +\infty} \delta(r)/r = 0$ such that for every $r \ge r_0$ the map $f_r$ is a $(1 + \varepsilon, \delta(r))$-quasi isometry from $(S_1(r), d_1)$ to $(S_2(r), d_2)$.

*Proof.* Fix a positive number $\varepsilon$. By assumption, there exists a positive number $b$ such that the distances $d_1$ and $d_2$ are $(1 + \varepsilon, b)$-quasi isometric through the identity map. Take a positive number $r_0 > (1 + \varepsilon)b$, and $r \ge r_0$. Let $x$ be a point in $S_1(r)$. Since $d_1(o, x) = r$, we have

$$(1 + \varepsilon)^{-1}r - b \le d_2(o, x) \le (1 + \varepsilon)r + b.$$

This implies

$$(4) \qquad d_2(f_r(x), x) = \frac{|r - d_2(o, x)|}{d_2(o, x)} \le \frac{\varepsilon r + b}{(1 + \varepsilon)^{-1}r - b}.$$

Denote by $\delta(r)$ the number $\delta(r) := b+2\frac{\varepsilon r + b}{(1 + \varepsilon)^{-1}r - b}$. By triangular inequality, for every $x, y$ in $S_1(r)$,

$$d_2(x, y) - d_2(f_r(x), x) - d_2(f_r(y), y) \le d_2(f_r(x), f_r(y)) \le d_2(f_r(x), x) + d_2(x, y) + d_2(y, f_r(y)).$$

Hence, using the quasi isometric inequality and Equation (4), we obtain

$$(1 + \varepsilon)^{-1}d_1(x, y) - \delta(r) \le d_2(f_r(x), f_r(y)) \le (1 + \varepsilon)d_1(x, y) + \delta(r),$$

for every $x, y$ in $S_1(r)$. Hence $f_r$ is a $((1 + \varepsilon), \delta(r))$-quasi isometry from $(S_1(r), d_1)$ to $(S_2(r), d_2)$, and $\frac{\delta(r)}{r}$ tends to 0 when $r$ goes to $+\infty$. $\square$

We are now ready to prove the Corollary.

*Proof of Corollary A.2.2.* Suppose that $(\Sigma(\infty), Td_1)$ is compact. Then the sequence of metric spaces $(S_1(r), d_1/r)$ converges to $(\Sigma(\infty), l_1)$. Hence there is an integer $K$ and a positive number $r_1$ such that for every $r > r_1$, the metric space $(S_1(r), d_1/r)$ admits a 1-net with $K$ elements. We can now map this net to $(S_2(r), d_2/r)$ via the map $f_r$. We obtain that for every $r > \max(r_0, r_1)$, the metric space $(S_2(r), d_2/r)$ admits a $(1 + \varepsilon(r))$-net $\{\xi_1(r), \dots, \xi_K(r)\}$, where $\varepsilon(r)$ tends to 0 when $r$ goes to $+\infty$.

Now, up to extract, and making $r$ go to $+\infty$, we obtain a 1-net $\{\xi_1, \dots, \xi_K\}$ in $(\Sigma(\infty), l_2)$. By Theorem A.2.1 the metric space $(\Sigma(\infty), Td_2)$ is compact.

Moreover, if we suppose that the two Tits boundaries $(\Sigma(\infty), Td_1)$ and $(\Sigma(\infty), Td_2)$ are compact, then looking at nets equidistributed along the circles $(S_1(r), d_1/r)$ we
can show that for every positive number $\epsilon$ the perimeters $P_1$ and $P_2$ of $(\Sigma(\infty), Td_1)$ and $(\Sigma(\infty), Td_2)$ respectively, are related by

$$ (1 + \epsilon)P_1 \le P_2 \le (1 + \epsilon)P_1 . $$

Hence $(\Sigma(\infty), Td_1)$ and $(\Sigma(\infty), Td_2)$ are isometric, being isometric to a circle of same perimeter. $\square$

**Remark A.2.4.** Corollary A.2.2 gives a partial converse to [Oht96, Theorem 2], where Ohtsuka shows that given two Hadamard manifolds with isometric Tits boundaries and same growth rate, they are $(1 + \epsilon, b_\epsilon)$-quasi isometries between these two manifolds for every positive number $\epsilon$.

**A.3. Formula for the total curvature.** The following lemma, due to Ohtsuka [Oht88], strongly relates the Tits metric at infinity with the total curvature of a surface. Let $(\Sigma, g)$ be a Hadamard manifold of dimension 2.

We precise, given two ideal boundary points $\xi, \eta$ of $\Sigma$, the angular domain delimited by $\xi, \eta$ and a point $x$ of $\Sigma$. Given $x$ in $M$ denote by $c_\xi$ and $c_\eta$ the geodesic rays starting at $x$ with direction $\xi$ and $\eta$, respectively. One might want to define the angular domain determined by $\xi, \eta$ and $x$ by

$$ \{\sigma_t([0, 1]) \mid \sigma_t : [0, 1] \to \Sigma \text{ geodesic ray from } c_\xi(t) \text{ to } c_\eta(t), t \in [0, +\infty)\} . $$

This is not totally satisfying. Indeed, by A.1.9, if $Td(\xi, \eta) > \pi$, the above set is an infinite triangle with two ideal points. But if $Td(\xi, \eta) < \pi$, the above set is what we want. Also, we will cut the domain in several subdomains and we need to make a consistent choice, unambiguous when cutting the domain. Hence we make the following definition, following [Oht88].

**Definition A.3.1.** Let $\xi, \eta$ be two ideal boundary points of $\Sigma$.

If $Td(\xi, \eta) < +\infty$, the angular domain delimited by $\xi, \eta$ and $x$ is defined as being the connected component of $\Sigma \setminus (c_\xi([0, +\infty]) \cup c_\eta([0, +\infty]))$ containing the ray from $x$ to a midpoint between $\xi$ and $\eta$ in $(\Sigma(\infty), Td)$.

If $Td(\xi, \eta) = +\infty$, the angular domain delimited by $\xi, \eta$ and $x$ is any connected component of $\Sigma \setminus (c_\xi([0, +\infty]) \cup c_\eta([0, +\infty]))$.

**Lemma A.3.2 (Ohtsuka).** Let $(\Sigma, g)$ be a Hadamard manifold of dimension 2. Take $\xi$ and $\eta$ two ideal points, $x$ in $\Sigma$ and denote by $A$ the angular domain determined by $\xi, \eta$ and $x$. Denote by $K_y$ the Gaussian curvature of $\Sigma$ at $y$ in $\Sigma$. If $Td(\xi, \eta) < +\infty$, then

$$ \int_A K_y dv_g(y) = \mathcal{H}_x(\xi, \eta) - Td(\xi, \eta) . $$

If $Td(\xi, \eta) = +\infty$, then

$$ \int_A K_y dv_g(y) = -\infty . $$

The lemma is a generalized version of the Gauss Bonnet formula. We recall the Gauss Bonnet formula in the case of a geodesic triangle. Let $(\Sigma, g)$ be a Riemannian 2-dimensional manifold. Let $c_1, c_2, c_3$ be three geodesics defined in $[0, 1]$ such that $c_i(1) = c_{i+1}(0)$ for $i$ in $\{1, 2, 3\}$ and the concatenation of the three curves bounds a region homeomorphic to a disc. Denote by $T$ the corresponding geodesic triangle (the compact region bounded by the three segments). Denote by $\theta_1, \theta_2, \theta_3$ the interior angles of the triangle. Then we have the Gauss-Bonnet formula:

$$ \int_T K_y dv_g(y) = -\pi + \sum_{i=1}^3 \theta_i . $$
The idea of the proof of Lemma A.3.2 is to take geodesic rays $c_\xi$ and $c_\eta$ starting at $x$ and to consider the triangle with vertices $x, c_\xi(s), c_\eta(t)$. The Gauss-Bonnet formula gives us the total curvature inside this triangle, depending on the angles at each vertex. Now making $s$ and $t$ go to infinity gives us the result, using properties of the angle distance, in the case $\angle(\xi, \eta) < \pi$. If $Td(\xi, \eta) > \pi$, then by A.1.10 we can find several ideal boundary points $\xi_1, \dots, \xi_k$ such that $Td(\xi_i, \xi_{i+1}) < \pi$ and this yields the result for $Td < +\infty$. When $Td = \infty$, we can divide the segment between $\xi$ and $\eta$ in $M(\infty)$ in an arbitrary large number of segments of Tits length bigger than $\pi$. The same argument applies to conclude that the total curvature cannot be bounded by below.

This beautiful formula implies a strong consequence about the total curvature of Riemannian surfaces with finite diameter Tits boundary.

**Theorem A.3.3.** Let $(\Sigma, g)$ be a Hadamard surface. The Tits distance in the ideal boundary $\Sigma(\infty)$ is that of a circle of perimeter $1 \le P < +\infty$ if and only if the total curvature of $(\Sigma, g)$ satisfies

$$ \int_{\Sigma} K = 2\pi - P . $$

*Proof.* Suppose that the Tits distance on the ideal boundary is that of a circle of perimeter $P$ in $[1, +\infty[$. For $k = [P/\pi] + 1$, there are $k$ points $\xi_1, \dots, \xi_k$ of $\Sigma(\infty)$ such that

* The distance $Td(\xi_i, \xi_{i+1})$ is less than $\pi$, for every $i$,

* The distance $Td(\xi_i, \xi_j)$ is bigger than $\pi$, for every $|i - j| > 1$,

* The sum of the distances $Td(\xi_i, \xi_{i+1})$, for $i$ through $\{1, \dots, k\}$ equals $P$.

Here the indices are modulo $k$. We can apply the formula of Ohtsuka, Lemma A.3.2, for a point $x$ in $\Sigma$ and the domains bounded by $\xi_i, \xi_{i+1}$ for $i$ through $\{1, \dots, n\}$. Now, the sum of the angles $\angle_x(\xi_i, \xi_{i+1})$ at $x$ equals $2\pi$, and the sum of the distances $\angle(\xi_i, \xi_{i+1}) = Td(\xi_i, \xi_{i+1})$ equals $P$, hence the formula for the total curvature.

Now if there are two points $\xi, \eta$ in the ideal boundary $\Sigma(\infty)$ such that $Td(\xi, \eta) = +\infty$, then the formula of Ohtsuka, Lemma A.3.2, says that the total curvature of $\Sigma$ is $-\infty$. $\square$

**Remark A.3.4.** The proof is done by finding a $\pi$-net in $(\Sigma(\infty), Td)$, describing the Tits distance completely.

## APPENDIX B. FLAT METRICS WITH CONICAL SINGULARITIES

In this appendix, we study the flat metric with conical singularities associated with a holomorphic quartic differential on the complex plane. We show that Hadamard theory applies and compute the Tits metric on the ideal boundary of such a metric. Our main references are [Tro86] for flat surfaces with conical singularities, [Str84] for quadratic differentials, and [Bal95] for Hadamard spaces. For this appendix, let $\varphi$ be a holomorphic quartic differential in the plane $\mathbb{C}$.

### B.0.1. Definition.
A holomorphic quartic differential $\varphi$ on $\mathbb{C}$ is written in the $z$-parameter

$$ \varphi = \varphi(z)dz^4, $$

where $z \mapsto \varphi(z)$ is a holomorphic function on $\mathbb{C}$.

It defines a smooth flat metric $g$ on $\mathbb{C}\{\text{ zeros of } \varphi\}$ with the following formula:

$$ g_z = |\varphi(z)|^{\frac{1}{2}} |dz|^2. $$

This formula is invariant by coordinate change.
In the neighborhood of a zero, it is possible ([Str84, Chapter III, Paragraph 6]) to find a local parameter $w$ such that the quartic differential $\varphi$ is written in this coordinate

$$ \varphi = \varphi(w)dw^4 = w^k dw^4, $$

where $k$ is the order of the zero. This parameter is unique up to the multiplication by a $(k+4)$-th root of unity. In this parameter, the metric $g$ is written

$$ g_w = |w|^{\frac{k}{2}} |dw|^2, $$

and is isometric to a Euclidean cone of angle $\theta = \frac{\pi}{2}k + 2\pi$ ([Tro86, Proposition 1]). At a nonzero point $z$, the metric $g$ has vanishing curvature. This is the reason why we talk about flat metrics with conical singularities.

B.0.2. *Curvature near a zero.* The angle of the cone being bigger than $2\pi$, the surface behaves locally like a non-positively curved manifold. Indeed, such a cone is the limit of smooth metrics on the plane with total curvature $2\pi - \theta = -\frac{\pi}{2}k$.

As a consequence the distance function associated with the metric $g$ is $\mathrm{CAT}(0)$ and the theory of Hadamard spaces applies.

**Remark B.0.1.** Flat surfaces with conical singularities are very close to Hadamard surfaces. They are manifolds, limits of Hadamard surfaces, and the metric is singular only in a discrete set (finite in the particular case of polynomial quartic differentials).

B.0.3. *Total curvature.* Suppose now that $\varphi$ is a polynomial quartic differential, of degree $N$. Then the total curvature of $(\mathbb{C}, g)$ equals

$$ \int_{\mathbb{C}} K = -\frac{\pi}{2} N. $$

B.1. **Ideal boundary.** We define the ideal boundary of $(\mathbb{C}, g)$ in the same way as for Hadamard manifolds.

**Definition B.1.1.** The ideal boundary of $(\mathbb{C}, g)$ is the set of equivalence classes of rays parametrized by arclength where two rays are equivalent if they stay at bounded distance.

The ideal boundary of $(\mathbb{C}, g)$ has the topological structure of a circle, but this does not come from an identification with the unit tangent space at a point $z \in \mathbb{C}$. Indeed, they can be several non-equivalent rays starting from the same point and being equal along an interval $[0, t]$, because of the zeros of the metric $g$ (Figure 7).

<figure><img src="image_19.png" /><figcaption>FIGURE 7. Picture of several non-equivalent rays starting from the same point $z$ and being equal between $z$ and a zero $z_0$ of $g$.</figcaption></figure>

Instead we can define a generalized exponential map $\exp_0$, in the case of finite degree, as follows. At a zero $z$ of order $k$, the small circles centered at $z$ have length $\frac{\pi}{2}k + 2\pi$. We parametrize such a small circle $C$ by arclength, with the origin being in the geodesic segment between 0 and $z$. The ray from 0 to $z$ can be continued as a ray for a choice of a point in $C$ between $\pi$ and $\frac{\pi}{2}k + \pi$. Hence, we define $\exp_0$ in a cone over a circle of perimeter $2\pi + \frac{\pi}{2}k$, and it is not injective for points with radius too small.
B.1.1. The l metric. In the same way as for Hadamard surfaces, the l-metric on the ideal boundary is defined by, given $\xi_1, \xi_2$ two points on the ideal boundary of $(\mathbb{C}, g)$ and $c_1, c_2$ being two rays representing the $\xi_i$'s and starting from the same point

$$l(\xi_1, \xi_2) = \lim_{t \to +\infty} \frac{d(c_1(t), c_2(t))}{t}.$$

This definition does not depend on the chosen starting point.

B.1.2. Computations a particular case. Suppose that the quartic differential $\varphi$ equals $\varphi(z)dz^4 = z^N dz^4$. We will compute the l-distance on the ideal boundary of $(\mathbb{C}, g)$. For $\theta$ in $[0, 2\pi]$ fixed, the curves

$$c_{\theta}: [0, +\infty[ \to \mathbb{C} \\ t \mapsto \exp(i\theta) \left( \frac{N+4}{4} t \right)^{\frac{4}{N+4}}$$

are geodesic rays parametrized by arc length.

The g-geodesic circle of radius $r$ is parametrized by

$$\gamma_r : [0, 2\pi] \to \mathbb{C} \\ \theta \mapsto \exp(i\theta) \left( \frac{N+4}{4} r \right)^{\frac{4}{N+4}},$$

and has perimeter

$$\int_{0}^{2\pi} g_{\gamma_r(\theta)}(\dot{\gamma}_r, \dot{\gamma}_r)^{1/2} d\theta = \frac{\pi}{2}(N+4)r.$$

We obtain easily that the Tits boundary of $(\mathbb{C}, g)$ is a circle of perimeter $\frac{\pi}{2}(N+4)$.

B.1.3. Computations in the general case. Suppose now that the quartic differential $\varphi$ is a general polynomial of degree $N$, equal to

$$\varphi(z)dz^4 = (z^N + a_{N-1}z^{N-1} + \dots + a_1z + a_0)dz^4$$

in the z-coordinate. Denote by $g$ the induced metric with conical singularities.

**Proposition B.1.2.** For every positive number $\varepsilon$ there is a nonnegative number $b$ such that the identity is a $(1 + \varepsilon, b)$-quasi-isometry between $g$ and $|z|^{N/2}|dz|^2$.

*Proof.* Let $\varepsilon$ be a positive number, and take a positive number $R$ such that

$$\frac{|a_{N-1}|}{R} + \dots + \frac{|a_0|}{R^N} \le \varepsilon.$$

Denote by $K$ the set of points $z$ such that $|z| \le R$. By triangular inequality, we have for every $z$ in $\mathbb{C} \setminus K$

$$|z|^N(1-\varepsilon) \le |z|^N + a_{N-1}z^{N-1} + \dots + a_0 \le |z|^N(1+\varepsilon).$$

Hence $g$ and $|z|^{N/2}|dz|^2$ are $(1 + \varepsilon)$-biLipschitz in $\mathbb{C} \setminus K$. Denote by $d$ the distance function associated with $g$ and by $d_0$ the distance function associated with $|z|^{N/2}|dz|^2$. Define

$$b := \sup_{(x,y) \in K^2} \{|d(x,y) - d_0(x,y)|\}.$$

Then $d$ and $d_0$ are $(1 + \varepsilon, b)$-quasi-isometric on $\mathbb{C}$.

<figure><img src="image_20.png" /></figure>

We can now use Corollary A.2.2 to show the following.

**Proposition B.1.3.** The Tits boundary of $(\mathbb{C}, g)$ is a circle of perimeter $\frac{\pi}{2}(N+4)$.
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LABORATOIRE J.A. DIEUDONNÉ, UNIVERSITÉ CÔTE D'AZUR, FRANCE

Email address: amoriani@unice.fr