# A geometric description of the set of almost-fuchsian representations

Samuel Bronstein, Graham Andrew Smith

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Samuel Bronstein, Graham Andrew Smith. A geometric description of the set of almost-fuchsian representations. 2023. hal-03956297

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# A geometric description of the set of almost-fuchsian representations

SAMUEL BRONSTEIN, GRAHAM ANDREW SMITH

ABSTRACT. We prove that the set of proper minimal disks in the real hyperbolic 3-space whose principal curvatures belong in a closed set of $(-1, 1)$ is parametrized by a convex set of Hopf differentials. We give a quantitative estimate of this convex set in terms of sup norms of the quadratic differentials. Applying this result to disks equivariant under some surface group action, we get a parametrization of almost-fuchsian representations by a fiberwise convex open set in the cotangent bundle to Teichmüller space.

## CONTENTS

<table><tr><td>1. Context and Statement of Results</td><td>1</td></tr><tr><td>2. introduction and notation</td><td>2</td></tr><tr><td>3. Proof of main theorem</td><td>6</td></tr><tr><td>References</td><td>13</td></tr></table>

## 1. CONTEXT AND STATEMENT OF RESULTS

In this paper we study the geometry of the space of complete almost-fuchsian minimal disks in $\mathbb{H}^3$. In the special case of disks equivariant under the action of some cocompact surface group in $\mathbb{H}^3$, this naturally gives a description of the moduli space of almost-fuchsian representations by a fiberwise convex subset of the cotangent bundle of Teichmüller space. The same discussion yields a description of minimal almost-fuchsian disks invariant under some fixed group $\Gamma < \text{PSL}(2, \mathbb{R})$.

Let $\Sigma_g$ be a closed hyperbolic surface of genus $g \ge 2$ and let $\Gamma = \pi_1(\Sigma_g)$ denote its fundamental group. In her seminal work on minimal surfaces in hyperbolic 3-space [Uhl83], K. Uhlenbeck defines the notion of *almost-fuchsian representation*: it is a discrete and faithful representation $\rho$ of $\Gamma$ into $\text{PSL}(2, \mathbb{C})$ such that there is a $\rho$-equivariant immersion $f: \mathbb{H}^2 \to \mathbb{H}^3$ with minimal image and principal curvatures in $(-1, 1)$. Uhlenbeck showed that those representations are always convex-cocompact, and that they form an open neighborhood of the fuchsian locus inside $\text{Hom}(\Gamma, \text{PSL}(2, \mathbb{C}))$. In $\mathbb{H}^3$, convex-cocompact representations are quasi-fuchsian representations, hence the denomination of almost-fuchsian for this stronger notion.

Considering the minimal surface problem in $\mathbb{H}^3$, almost-fuchsian representations are of special interest because they admit a unique invariant minimal disk with small principal curvatures. This is not the case for arbitrary convex-cocompact representations, see for instance [HW15].

<footnote>Date: January 3, 2023.</footnote>
Krasnov–Schlenker [KS07] have extended the notion to almost-fuchsian immersions or almost-fuchsian disks, not necessarily invariant under a surface group. Seppi [Sep16] used it to show that quasi-circles with quasisymmetry constant close to 1 bound minimal disks.

We will work with the following notion of almost-fuchsian immersion.

**Definition 1.1.** An immersion $f: \mathbb{D} \to \mathbb{H}^3$ is said to be almost-fuchsian if it is proper, conformal with minimal image, and has principal curvatures in $(-1 + \epsilon, 1 - \epsilon)$, for some $\epsilon > 0$.

A subdisk of $\mathbb{H}^3$ is said to be an almost-fuchsian subdisk if it is the image of an almost-fuchsian immersion.

Let $\mathcal{Q}(\mathbb{D})$ denote the space of holomorphic quadratic differentials over $\mathbb{D}$. The main result we will prove in this paper is the following:

**Theorem 1.2.** Let $C \subseteq \mathcal{Q}$ denote the set of Hopf differentials of almost fuchsian immersions. Then

(1) 2 disks with same Hopf differentials are identified by an isometry of $\mathbb{H}^3$.

(2) $C$ is a bounded, open and convex subset of $\mathcal{Q}$, and

(3) $C$ contains all holomorphic quadratic differentials such that $|\phi|_{\infty} \le \frac{1}{2}$.

(4) For any $\phi \in C$, $|\phi|_{\infty} \le 1$.

Krasnov–Schlenker [KS07] already consider this parametrization for the case of holomorphic quadratic differentials invariant under a closed surface subgroup of $\mathrm{PSL}(2, \mathbb{C})$. In that case, it was already known that the set is open and star-shaped around 0, see for instance [Tra19].

The almost-fuchsian moduli space is an example of moduli space endowed with a hyperkähler structure, see [Don03] and [Hod05]. The hyperkähler structure actually coincides with the Feix-Kaleidin structure when considering $C$ as a neighborhood of the fuchsian moduli space, [Tra19].

This description of $C$ has 2 main consequences. First, it extends the notion of almost-fuchsian representation to any infinite discrete subgroup of $\mathrm{PSL}(2, \mathbb{R})$. Also, considering holomorphic quadratic differentials invariant under a surface group representation, we prove the following result.

**Theorem 1.3.** There exists a Finsler metric $p$ on the Teichmüller space $\mathcal{T}_g$ such that

$$ (1.1) \qquad \forall \phi \in T^*\mathcal{T}_g, p(\phi) < 1 \text{ if and only if } \phi \in C. $$

This Finsler metric is uniformly equivalent to the Finsler metric dual to the unit $L^{\infty}$ balls under the identification of the cotangent bundle with the space of holomorphic quadratic differentials.

The authors thank Nicolas Tholozan for his insights and discussions on this topic.

## 2. INTRODUCTION AND NOTATION

This section is a classical survey of the Gauss-Codazzi equation for almost-fuchsian disks. The results are well-known, we explain them here to make our notation clear. The reader comfortable with Gauss-Codazzi theory may go directly to the next section.

In the following, $\Sigma$ is a connected, oriented surface, with $\chi(\Sigma) < 0$. Let $(M, h)$ is a complete, 3-dimensional hyperbolic manifold, and let $f: \Sigma \to M$ be a smooth immersion.
Let $g_0$ denote the induced metric over $S$, let $\mathbb{I}$ denote the second fundamental form of $f$, and let $A$ denote its shape operator. Recall that $f$ is said to be **minimal** whenever its mean curvature $H := \text{Tr}(A)$ vanishes.

Let $\kappa$ denote the intrinsic curvature of $g_0$. The Gauss equation is

$$ (2.1) \qquad \kappa = \det_{g_0}(\mathbb{I}) - 1. $$

where the determinant of the second fundamental form $\mathbb{I}$ is taken with respect to $g_0$. The determinant of $\mathbb{I}$ is also the determinant of the shape operator $A$.

**Definition 2.1** (Hopf differential). The Hopf differential $\phi$ of $f$ is the $(2,0)$-component of $\mathbb{I}$. That is, $\phi$ is the symmetric bilinear form given by

$$ (2.2) \qquad \phi(\xi, \nu) := \frac{1}{4} \left( \mathbb{I}(\xi, \nu) - i \mathbb{I}(J\xi, \nu) - i \mathbb{I}(\xi, J\nu) - \mathbb{I}(J\xi, J\nu) \right). $$

In particular, when $f$ is minimal we readily obtain:

$$ (2.3) \qquad \det_{g_0}(\mathbb{I}) = -|\phi|_{g_0}^2. $$

In the following, we will denote $g$ a fixed hyperbolic metric on a Riemann surface, we call the pair $(u, \phi)$ Gauss-Codazzi data for the immersion, where $e^{2u}g$ is the induced metric and $\phi$ is the Hopf differential of the corresponding immersion.

**Lemma 2.2.** Let $(\Sigma, g_0)$ be a riemannian surface and let $g_1 = e^{2u}g_0$ be a metric conformal to $g_0$. Let $\Delta^i$ and $\kappa^i$ denote respectively the laplacian and curvature of the metric $g_i$ for $i \in \{0, 1\}$. Then

$$ (2.4) \qquad \kappa^1 e^{2u} = \kappa^0 - \Delta^0 u $$

$$ (2.5) \qquad \kappa^0 = (\kappa^1 + \Delta^1 u)e^{2u}. $$

*Proof.* Denote $\text{Ric}^i$ the Ricci tensor of $g_i$. The following equation in Besse's book, 1.159 d) [Bes07], relates the Ricci curvatures of $g_1$ and $g_0$:

$$ (2.6) \qquad \text{Ric}^1 = \text{Ric}^0 - (\Delta^0 u)g_0 $$

The remaining equations follow from the fact that, for a riemannian surface $(\Sigma, g_i)$, $\text{Ric}^{g_i} = \kappa^i g_i$. Note that Besse uses the analysts' laplacian, whose sign is the inverse of that of the geometer's laplacian. $\square$

Combining these equations yields the Gauss-Codazzi equation.

**Proposition 2.3** (Gauss-Codazzi). Let $f: \Sigma \to (M, h)$ be a minimal immersion of a surface into a hyperbolic manifold. Let $g_0$ denote the induced metric and let $g = e^{-2u}g_0$ denote the unique, complete hyperbolic metric in its conformal class. Then $u$ satisfies

$$ (2.7) \qquad \Delta u = -1 + e^{2u} + e^{-2u} |\phi|_g^2. $$

*Proof.* By lemma 2.2,

$$ (2.8) \qquad \kappa_0 = (-1 - \Delta u)e^{-2u}. $$

By Gauss equation,

$$ (2.9) \qquad \kappa_0 = -1 - e^{-4u} |\phi_0|^2_g. $$

The result follows upon combining these two equations. $\square$
By the fundamental theorem of surface theory, for every pair $(u, \phi)$, where $u : \Sigma \to \mathbb{R}$ and $\phi$ is a holomorphic quadratic differential on $\Sigma$ satisfying the Gauss-Codazzi equation (2.7), there exists an immersion of $\tilde{\Sigma}$ into $\mathbb{H}^3$ having this Gauss-Codazzi data. As the surface considered is hyperbolic $\tilde{\Sigma}$ identifies with the open disk $\mathbb{D}$, and the immersion is equivariant with regard to some representation $\rho : \pi_1\Sigma \to \text{Isom}(\mathbb{H}^3)$. Furthermore, this immersion is unique up to post composition by an isometry of $\mathbb{H}^3$.

We will require the following result of Li-Mochizuki [LM20].

**Proposition 2.4** (Sub-supersolution method). Let $(\Sigma, g)$ be a riemannian surface, and let $F$ be a smooth function defined on $\Sigma \times \mathbb{R}$. Consider the equation (E):

$$ (2.10) \qquad \Delta_g u = F(x, u) $$

Assume there are $u_-, u_+ : \Sigma \to \mathbb{R}$ satisfying:

$$ (2.11) \qquad u_- < u_+ $$

$$ (2.12) \qquad \Delta_g u_- \geq F(x, u_-) $$

$$ (2.13) \qquad \Delta_g u_+ \leq F(x, u_+) $$

Then the equation (E) admits at least one solution $u$ satisfying $u_- \leq u \leq u_+$.

First remark that almost-fuchsian subsurfaces of $\mathbb{H}^3$ are necessarily uniformized by the Poincaré disk, and that the conformal factor between the induced metric and the hyperbolic one is bounded:

**Lemma 2.5.** Let $M$ be a complete hyperbolic 3-manifold. Let $f : \Sigma \to M$ be an almost-fuchsian immersion, then its induced metric $g_0$ is conformal to a complete hyperbolic metric $g$. Denote $g_0 = e^{2u}g$. Then $u$ satisfies.

$$ (2.14) \qquad -\frac{\ln 2}{2} \leq u \leq 0. $$

*Proof.* Consider the prescribed curvature equation on the universal cover :

$$ (2.15) \qquad \Delta^{g_0} v = e^{2v} - \kappa_{g_0}. $$

For any solution $v$, the corresponding metric $e^{2v}h_0$ will be hyperbolic. As $f$ is almost fuchsian, the curvature $\kappa_{h_0}$ satisfies $-2 \leq \kappa_{h_0} \leq -1$. Upon applying the sub-supersolution method (Prop 2.4) to (2.15) with subsolution $v_- = 0$ and supersolution $v_+ = \frac{\ln 2}{2}$, we get a solution $v$ satisfying

$$ (2.16) \qquad 0 \leq v \leq \frac{\ln 2}{2}. $$

Since $v$ is bounded, $(\Sigma, e^{2v}g_0)$ is quasi-isometric to $(\Sigma, g_0)$. As $f$ is almost-fuchsian, it is proper. Thus $g_0$ is complete, and $e^{2v}g_0$ is also complete. By the Ahlfors-Schwarz-Pick lemma [Ahl38], the complete hyperbolic metric. is unique in its conformal class, hence $g = e^{2v}g_0$ is complete hyperbolic. Denote $u = -v$, we check:

$$ (2.17) \qquad -\frac{\ln 2}{2} \leq u \leq 0, $$

As asserted.

□

*Remark.*
* With the same proof, one can prove that this is a general property of complete pinched Hadamard disks, due to Bland and Kalka [BK86]. It is also stated in Benoist-Hulin [BH20], Prop. 3.1. : A complete pinched hadamard disk is uniformly quasiconformal to the complete hyperbolic disk.

* In [AS20], Alvarez and Smith show a more general version of this: The conformal factor is actually bounded in $C^{k+2,\alpha}$.

**Theorem 2.6.** Let $g$ be a complete hyperbolic metric on $\Sigma$, $u : \Sigma \to \mathbb{R}$ a bounded continuous map, and $\phi$ a holomorphic quadratic differential on $\Sigma$ satisfying

$$ (2.18) \qquad \Delta^g u = -1 + e^{2u} + e^{-2u} |\phi|^2, \quad \text{and} $$

$$ (2.19) \qquad \sup_{z \in \Sigma} (e^{-2u(z)} |\phi(z)|) < 1. $$

Then there is an almost-fuchsian immersion of $\Sigma$ into a hyperbolic manifold such that $e^{2u}g$ is the induced metric and $\phi$ is the (2,0)-part of the second fundamental form. Conversely, for any almost-fuchsian immersion the associated data satisfies these relations.

*Proof.* By the fundamental theorem of surface theory, we get an immersion into a hyperbolic manifold. Since $u$ is bounded, the induced metric is complete and it follows from (2.19) that the principal curvatures lie in $(-1,1)$. Hence $f$ is almost-fuchsian.

Conversely, if $f$ is almost-fuchsian, consider $(u, \phi)$ the associated Gauss-Codazzi data. Then Gauss-Codazzi is verified, the bound on the principal curvatures yields (2.19) and the boundedness of $u$ follows from the previous lemma. $\square$

Note that almost-fuchsian subdisks of $\mathbb{H}^3$ are always properly embedded and bound a quasicircle at infinity. Indeed, this readily follows from the original work [Uhl83] of Uhlenbeck and the argument of Theorem 1.3 of [HW13]. For the reader's convenience, we state and prove this result here.

**Theorem 2.7.** Let $e: \mathbb{D} \to \mathbb{H}^3$ be an almost-fuchsian disk. Then the disk $e(\mathbb{D})$ is properly embedded and bounds a quasi-circle at infinity. Furthermore, if $k$ denotes the supremum of the principal curvatures of $e$, the quasisymmetry constant of the quasi-circle is bounded by $\frac{1+k}{1-k}$.

*Proof.* Let $g$ be the Poincaré metric of $\mathbb{D}$. Let $g_0 = e^{2u}g$ denote the induced metric of $e$ on the disk $\mathbb{D}$. Consider $\nu$ the unit normal vector field, and define $\tilde{e}$ by

$$ (2.20) \qquad \tilde{e} : \begin{cases} \mathbb{D} \times \mathbb{R} \to \mathbb{H}^3, \text{ and} \\ (x, t) \mapsto \exp_x(t\nu_x). \end{cases} $$

where $\exp$ denotes the exponential map of $\mathbb{H}^3$. Let $A$ denote the shape operator of $f$. The same computation as of lemma 2.1. of [HW13] gives the following expression of the metric induced by $\tilde{e}$:

$$ (2.21) \qquad h^{\nu}(x, t) := h_t(x) \oplus dt^2, $$

where, for all $t$,

$$ (2.22) \qquad h_t(\cdot, \cdot) = e^{2u} g((\cosh(t)I_0 + \sinh(t)A)\cdot, (\cosh(t)I_0 + \sinh(t)A)\cdot). $$

Consider $0 < k < 1$ such that the operator norm of $A$ is less than $k$. Since $|A| \le k < 1$, $h^{\nu}$ is complete and thus $\tilde{e}$ is a diffeomorphism. In particular, $e$ is properly embedded.
Let $f: \mathbb{D} \to \mathbb{H}^3$ be an isometric parametrization of a totally geodesic plane, let $v_f$ denote its unit normal vector field, and define $\tilde{f}: \mathbb{D} \times \mathbb{R} \to \mathbb{H}^3$ by

$$ (2.23) \qquad \tilde{f}(x, t) = \exp_{f(x)}(t\nu_f(x)). $$

The metric induced by $\tilde{f}$ is

$$ (2.24) \qquad \tilde{g}(x, t) = \cosh^2(t)g \oplus dt^2. $$

It follows from (2.24) and (2.22) that $\alpha := e \circ f^{-1}$ is bilipschitz. It therefore extends (see Gehring [Geh62] or Mostow [Mos68]) to a quasiconformal map $\tilde{\alpha}$ from $\partial_{\infty}\mathbb{H}^3$ to itself. It follows that $\Gamma := \partial_{\infty}e(\mathbb{D})$ is a quasicircle.

Note that, for all $t$, the restrictions of $h^0$ and $\tilde{g}$ to the slice $\mathbb{D} \times \{t\}$ are $K$-quasiconformal, with

$$ (2.25) \qquad K = \frac{1+k}{1-k}. $$

Letting $t$ tend to $+\infty$, it follows that $\tilde{\alpha}$ is $K$-quasiconformal. As the boundary of the disk is the image by $\tilde{\alpha}$ of an equator in $\mathbb{S}^2$, this completes the proof. $\square$

### 3. PROOF OF MAIN THEOREM

As in this paper we are mainly interested in the description of conformal almost-fuchsian disks, let us define the space we want to describe:

**Definition 3.1.** We denote $\mathcal{AF}$ the set of almost-fuchsian immersions $\mathbb{D}^2 \to \mathbb{H}^3$ up to postcomposition by an isometry of $\mathbb{H}^3$.

The set $\mathcal{AF}$, equipped with its quotient topology, can be identified with the set of almost-fuchsian immersions with prescribed image and differential of the origin $o$. As such, it can be viewed as a subspace of the space of immersions from the disk into $\mathbb{H}^3$.

*Remark.* The space $\mathcal{AF}$ does not parametrize almost-fuchsian subdisks of $\mathbb{H}^3$. However, the set of almost-fuchsian subdisks up to isometry can be identified to the quotient of $\mathcal{AF}$ by a $\mathrm{PSL}(2, \mathbb{R})$ action by precomposition, which is not a properly discontinuous action.

In order to prove our theorems in a noncompact setting, we will use the following version of Omori-Yau's theorem [Omo67, Yau75]. It can be found in Li-Mochizuki [LM20], we reproduce it here for the reader's convenience.

**Theorem 3.2 (Omori-Yau).** Let $(M, h)$ be a complete manifold with Ricci curvature bounded from below. Let $u: M \to \mathbb{R}$ be a map bounded from above and let $\varepsilon > 0$. Then there is $x \in M$ such that

$$ (3.1) \qquad u(x) \geq \sup u - \varepsilon, $$

$$ (3.2) \qquad |\nabla_h u(x)| \leq \varepsilon, \quad \text{and} $$

$$ (3.3) \qquad \Delta_h u(x) \leq \varepsilon. $$

The first step is to show that the moduli space of almost-fuchsian disks injects into the space of bounded holomorphic quadratic differentials.
**Proposition 3.3.** Let $\phi$ be a holomorphic quadratic differential on the disk. Then there is at most one bounded solution to

$$ (3.4) \qquad \Delta u = -1 + e^{2u} + e^{-2u} |\phi|^2 $$

such that $e^{-2u}|\phi| < 1$ everywhere.

*Proof.* Suppose $(u, \phi)$ and $(v, \phi)$ are two bounded solutions of (3.4) such that $e^{-2u}|\phi| < 1$ and $e^{-2v}|\phi| < 1$. Then $u - v$ is bounded and satisfies

$$ (3.5) \qquad \Delta(u-v) = (e^{2v} - e^{-2v}|\phi|^2)(e^{2(u-v)} - 1) + |\phi|^2 e^{-2v}(e^{2(u-v)} + e^{2(v-u)} - 2). $$

Using the positivity of the second term of the right hand side and the almost-fuchsian property of $v$, one gets

$$ (3.6) \qquad \Delta(u-v) \geq (e^{2v} - e^{-2v}|\phi|^2)(u-v). $$

Applying Omori-Yau's maximum principle gives $u - v \leq 0$. Exchanging $u$ and $v$, one has $u = v$, thus the corresponding almost-fuchsian disks are conjugated. $\square$

*Remark.* The same reasoning actually shows that if $(e^{2u}g, \phi)$ is almost-fuchsian, then any upper bounded $v$ solution to the Gauss-Codazzi equation with quadratic differential $\phi$ must satisfy:

$$ (3.7) \qquad v \leq u. $$

In [HLT21], Huang, Lucia and Tarantello study the solutions of the Gauss-Codazzi equation on closed surfaces, of which the maximality of almost-fuchsian solutions is a consequence.

Moreover, we will show that the image of the almost-fuchsian locus in the space of bounded holomorphic quadratic differentials on the disk is actually open. Combining Proposition 3.3 with the fundamental theorem of surface theory, we get:

**Corollary 3.4.** For $f: \mathbb{D}^2 \to \mathbb{H}^3$ an immersion, denote $\phi_f$ its Hopf differential. Then the map:

$$ (3.8) \qquad \Phi: \left\{ \begin{array}{rcl} \mathcal{A}\mathcal{F} & \to & QD(\mathbb{D}^2) \\ f & \mapsto & \phi_f \end{array} \right. $$

is injective.

*Remark.* The statement of this corollary in the equivariant case is the Lemma 2.11 of Krasnov-Schlenker [KS07].

*Proof.* We already know that the Gauss-Codazzi data of an immersion characterizes it up to isometry. The prop (3.3) shows that if two almost-fuchsian immersions have same Hopf differentials, then they have the same Gauss-Codazzi data, hence the injectivity of our map. $\square$

**Theorem 3.5.** The image of the almost-fuchsian locus is open in the space of bounded holomorphic quadratic differentials.

*Proof.* Consider $\phi$ in $\mathfrak{S}(\mathcal{A}\mathcal{F})$, $u$ and $0 < \delta < 1$ satisfying

$$ (3.9) \qquad \Delta u = -1 + e^{2u} + e^{-2u} |\phi|^2 \quad \text{and} $$

$$ (3.10) \qquad \sup_{z \in \mathbb{D}} e^{-2u} |\phi| \leq 1 - \delta. $$
We will prove that the linearized operator of (3.9) $L_u : v \mapsto -\Delta v + 2(e^{2u} - e^{-2u}|\phi|^2)v$ is an isomorphism between the Banach spaces $C^{2,\alpha}(\mathbb{D})$ and $C^{0,\alpha}(\mathbb{D})$, where here $\mathbb{D}$ is equipped with its Poincaré metric, hence, using the local inversion principle, we will be able to find a solution $\tilde{u}$ uniformly close to $u$ on a neighborhoof of $\phi$, which then has to be almost-fuchsian.

At first, as $u$ is bounded, $L_u$ trivially defines a bounded linear map from $C^{2,\alpha}(\mathbb{D})$ towards $C^{0,\alpha}(\mathbb{D})$. The almost-fuchsian condition implies

$$ (3.11) \qquad \frac{\delta}{2} \le e^{2u} - e^{-2u}|\phi|^2 \le 1. $$

Hence $L_u$ is strictly elliptic. We now show that $L_u$ is injective. Indeed, choose $v \in \ker L_u$. Since $v$ is bounded, Omori-Yau's maximum principle implies

$$ (3.12) \qquad v \le 0. $$

The same reasoning to $-v$ shows that $v = 0$, hence $L_u$ is injective, as asserted.

It remains only to prove surjectivity. First consider a compact set $K$ in $\mathbb{D}$ with smooth boundary. It is a standard consequence of elliptic theory and of the strict ellipticity of $L_u$ that $L_u$ is an isomorphism from $C_0^{2,\alpha}(K)$ into $C^{0,\alpha}(K)$. Now choose $u \in C^{0,\alpha}(\mathbb{D})$, and a compact exhaustion $(K_n)$ of $\mathbb{D}$ such that each $K_i$ has smooth boundary. For each $i$ there is a unique $v_i \in C_0^{2,\alpha}(K_i)$ such that $L_u v_i = w \mathbf{1}_{K_i}$. We claim that the sequence $(|v_n|_\infty)$ is bounded. Indeed, by the maximum principle, for all $i$,

$$ (3.13) \qquad |u_i|_\infty \le \max\{2\delta, 1\} |w \mathbf{1}_{K_i}|_\infty \le 2|w|_\infty. $$

Fix a radius $r > 0$ and $x \in \mathbb{D}$. For $i$ big enough such that $B(x, 2r) \subset K_i$,

$$ (3.14) \qquad |L_u v_i|_{C^{0,\alpha}(B(x,2r))} \le |w|_{C^{0,\alpha}}, \text{ and} $$

$$ (3.15) \qquad |v_i|_{C^0(B_{x,2r})} \le 2|w|_{C^0}. $$

Hence, by the Schauder estimates, there is a $C > 0$ depending only on $r$ such that

$$ (3.16) \qquad |v_i|_{C^{2,\alpha}(B(r,x))} \le C|w|_{C^{0,\alpha}}. $$

By the Arzela-Ascoli theorem, up to extracting a subsequence, we can assume there exists $v_\infty \in C_{\text{loc}}^{2,\alpha}$ such that $v_n \to v_\infty$ on every compact set in the $C^{2,\beta}$ topology for $\beta < \alpha$. Furthermore

$$ |v_\infty|_{C^{2,\alpha}} \le C' \sup_{x \in \mathbb{H}^2} |v_\infty|_{C^{2,\alpha}(B(r,x))} $$

$$ \le C' \sup_{x \in \mathbb{H}^2} \limsup_{i \to \infty} |v_i|_{C^{2,\alpha}(B(r,x))} $$

$$ \le C'C|w|_{C^{0,\alpha}}, $$

so that $v_\infty \in C^{2,\alpha}(\mathbb{H}^2)$. By continuity of $L_u$,

$$ (3.17) \qquad L_u v_\infty = w. $$

Thus $L_u$ is surjective.

By the Closed Graph Theorem, $L_u$ is a linear isomorphism. It follows by the inverse function theorem that the image of the almost-fuchsian locus is open in the space of bounded quadratic differentials, as desired. $\square$

Equipped with this theorem, we will consider a perturbation $\phi_t$ of $\phi$ in the image of the almost-fuchsian locus and we will consider the family $(u_t)$ of bounded maps satisfying the Gauss-Codazzi equation. In particular, we will need the derivatives $\dot{u}_t$ and $\ddot{u}_t$ to be bounded.
The following is actually a straightforward consequence of the implicit function theorem we used.

**Proposition 3.6.** Consider $(e^{2u}, \phi)$ an almost-fuchsian disk. Let $f_t$ be a $C^{0,\alpha}$ perturbation of $|\phi|^2$. Denote by $(u_t)$ the smooth branch of solutions to :

$$ (3.18) \qquad \Delta u_t = -1 + e^{2u_t} + e^{-2u_t} f_t $$

for $t$ in a neighborhood of 0. Then $\dot{u}_t$ and $\ddot{u}_t$ are bounded.

*Proof.* Consider $F: C^{2,\alpha}(\mathbb{D}) \times [0, 1] \to C^{0,\alpha}(\mathbb{D})$ defined by $F(v, t) = \Delta v + 1 - e^{2v} - e^{-2v}f_t$. Thanks to the previous lemma, we know the local existence of a smooth $G: [0, 1] \to C^{2,\alpha}(\mathbb{D})$ such that $F(G(t), t) = 0$. It follows by smoothness of $G$ that $\dot{u}_t, \ddot{u}_t \in C^{2,\alpha}(\mathbb{D})$ for all $t$, and the result follows. $\square$

We are now able to show that the almost-fuchsian locus retracts to the set of almost-fuchsian immersions with zero quadratic differentials.

**Theorem 3.7.** Let $\phi \in \mathfrak{F}(\mathcal{A}\mathcal{F})$ and let $\psi$ be a holomorphic quadratic differential on the disk satisfying

$$ (3.19) \qquad \forall z \in \mathbb{D}, |\psi(z)| \le |\phi(z)|. $$

Then $\psi \in \mathfrak{F}(\mathcal{A}\mathcal{F})$.

*Proof.* Let $u: \mathbb{D} \to \mathbb{R}$ be such that $(u, \phi)$ is the Gauss-Codazzi data of an almost-fuchsian disk. Let $f_t: \mathbb{D} \to [0, \infty[$ be a smooth map such that $f_0 = |\psi|^2$ and $f_1 = |\phi|^2$. As $|\psi| \le |\phi|$, one can choose $f$ so that the maps $t \mapsto f_t(z)$ are nondecreasing.

We will show that $\psi \in \mathfrak{F}(\mathcal{A}\mathcal{F})$, i.e. that there is a solution $v$ to

$$ (3.20) \qquad \Delta v = -1 + e^{2v} + e^{-2v}f_0, \text{ and} $$

$$ (3.21) \qquad \sup_{z \in \mathbb{D}} e^{-4v}f_0 < 1. $$

To do so, consider for $t \in [0, 1]$ the equation.

$$ (3.22) \qquad \Delta u_t = -1 + e^{2u_t} + e^{-2u_t} f_t. $$

Let $I \subseteq [0, 1]$ denote the set of all $s$ such that, for all $t \ge s$, this equation has a bounded solution $u_s$ satisfying $|e^{-4u_s}f_s|_{\infty} < 1$, for some $\epsilon > 0$. By Theorem 2.6, $1 \in I$. By Theorem 3.5 applied to a $C^{0,\alpha}$ perturbation, $I$ is open. It thus remains only to show that $I$ is closed.

At first, consider $t_0 = \inf I$ and $(t_n) \in I$ a monotonous sequence such that $t_n \to t_0$. By Lemma 2.5, the sequence $(u_{t_n})$ is uniformly bounded, from which it follows that $(\Delta u_{t_n})$ is also uniformly bounded. By Schauder's estimates, the sequence $(u_{t_n})$ is uniformly bounded in $C^{1,\alpha}$ on any compact set. Hence we may assume that $u_{t_n}$ converges towards a map $u_{t_0} \in C^{2,\alpha}(\mathbb{H})$ in the $C^{2,\beta}$ topology for all $\beta < \alpha$. Trivially $u_{t_0}$ solves (3.22).

It remains only to check that $|e^{-4u_{t_0}}f_{t_0}|_{\infty} < 1$. Differentiating the above equation with respect to $t$ yields,

$$ (3.23) \qquad \Delta \dot{u}_t = 2(e^{2u_t} - e^{-2u_t}f_t)\dot{u}_t + e^{-2u_t}\dot{f}_t. $$

By Lemma 2.5, there exists $B > 0$ such that $|u_t|_{\infty} < B$ for all $t$. It follows by the Omori-Yau maximum principle that, over $I$, $\dot{u}_t \le 0$, so that, for all $t \in I$,

$$ (3.24) \qquad |e^{-4u_t}f_t|_{\infty} \le |e^{-4u_1}f_1|_{\infty} < 1. $$
Closedness of $I$ now follows so that, by connectedness, $0 \in I$. At $t = 0$, this means that $(u_0, \psi)$ is the Gauss-Codazzi data of an almost-fuchsian disk. We have shown that $\psi \in \mathfrak{I}(\mathcal{A}\mathcal{F})$, as desired. $\square$

**Corollary 3.8.** Let $\phi \in \mathfrak{I}(\mathcal{A}\mathcal{F})$ and let $u : \mathbb{D} \to \mathbb{R}$ be the associated bounded map such that

$$ (3.25) \qquad \Delta u = -1 + e^{2u} + e^{-2u} |\phi|^2, \quad \text{and} $$

$$ (3.26) \qquad \sup_{z \in \mathbb{D}} e^{-2u} |\phi| \le 1 - \delta. $$

Denote $\phi_t = t\phi$ for $t \in [0, 1]$. Then for any $t \in [0, 1]$, there is a $u_t : \Sigma \to \mathbb{R}$ bounded such that $(e^{2u_t}g, \phi_t)$ satisfies the Gauss-Codazzi equation. Furthermore, for all $t \in [0, 1]$, $(e^{2u_t}g, \phi_t)$ is almost-fuchsian.

*Remark.* For the equivariant case, this computation can already be found in a survey of Trautwein [Tra19], step 3 of proof of theorem 5.13.

**Corollary 3.9.** *The moduli space of almost-fuchsian disks up to isometry is contractible. The set of almost-fuchsian immersions retracts onto the set of geodesic immersions of disks in $\mathbb{H}^3$.*

We continue with the proof of our main theorem:

**Theorem 3.10 (Convexity).** Let $(e^{2u_0}g, \phi_0)$ and $(e^{2u_1}g, \phi_1)$ be two pairs corresponding to almost-fuchsian disks. For $t \in [0, 1]$, denote $\phi_t := (1-t)\phi_0 + t\phi_1$. Then for any $t$, there is $(e^{2w_t}g, \phi_t)$ satisfying the Gauss-Codazzi equation, and it corresponds to an almost-fuchsian disk.

*Proof.* First, we write down Gauss-Codazzi and we differentiate it twice in $t$

$$ (3.27) \qquad \Delta w_t = -1 + e^{2w_t} + e^{-2w_t} |\phi_t|^2, $$

$$ (3.28) \qquad \Delta \dot{w}_t = 2\dot{w}_t e^{2w_t} - 2\dot{w}_t e^{-2w_t} |\phi_t|^2 + 2e^{-2w_t} g(\phi_t, \phi_1 - \phi_0), \quad \text{and} $$

$$ (3.29) \qquad \begin{aligned} \Delta \ddot{w}_t = 2\ddot{w}_t(e^{2w_t} - e^{-2w_t}|\phi_t|^2) &+ 4\dot{w}_t^2(e^{2w_t} + e^{-2w_t}|\phi_t|^2) \\ &-8\dot{w}_t e^{-2w_t} g(\phi_t, \phi_1 - \phi_0) + 2e^{-2w_t}|\phi_1 - \phi_0|^2. \end{aligned} $$

We first show that as long as $(e^{2w_t}g, \phi_t)$ is almost-fuchsian,

$$ (3.30) \qquad \ddot{w}_t + 4\dot{w}_t^2 \le 0. $$

This is a consequence of the Omori-Yau maximum principle and the following standard formula.

$$ (3.31) \qquad \Delta(\dot{w}_t^2) = 2\dot{w}_t \Delta \dot{w}_t + 2|\nabla \dot{w}_t|^2 \ge 2\dot{w}_t \Delta \dot{w}_t. $$

Then, using equations (3.28) and (3.29), we obtain

$$ (3.32) \qquad \begin{aligned} \Delta(\ddot{w}_t + 4\dot{w}_t^2) \ge & 2\ddot{w}_t(e^{2w_t} - e^{-2w_t}|\phi_t|^2) + 20\dot{w}_t^2 e^{2w_t} - 12\dot{w}_t^2 e^{-2w_t}|\phi_t|^2 \\ & +8\dot{w}_t e^{-2w_t} g(\phi_t, \phi_1 - \phi_0) + 2e^{-2w_t}|\phi_1 - \phi_0|^2. \end{aligned} $$

Judiciously rearranging the terms, we obtain

$$ (3.33) \qquad \begin{aligned} \Delta(\ddot{w}_t + 4\dot{w}_t^2) \ge & 2(\ddot{w}_t + 4\dot{w}_t^2)(e^{2w_t} - e^{-2w_t}|\phi_t|^2) + 12\dot{w}_t^2(e^{2w_t} - e^{-2w_t}|\phi_t|^2) \\ & +8\dot{w}_t^2 e^{-2w_t} |\phi_t|^2 + 8e^{-2w_t} g(\dot{w}_t\phi_t, \phi_1 - \phi_0) + 2e^{-2w_t} |\phi_1 - \phi_0|^2. \end{aligned} $$
As $8A^2 + 8AB + 2B^2 = 2(2A + B)^2$ is nonnegative, we obtain

$$ (3.34) \qquad \Delta(\ddot{w}_t + 4\dot{w}_t^2) \geq 2(\ddot{w}_t + 4\dot{w}_t^2)(e^{2w_t} - e^{-2w_t}|\phi_t|^2) + 12\dot{w}_t^2(e^{2w_t} - e^{-2w_t}|\phi_t|^2). $$

Hence, provided the pair is almost-fuchsian, we get

$$ (3.35) \qquad \Delta(\ddot{w}_t + 4\dot{w}_t^2) \geq 2(\ddot{w}_t + 4\dot{w}_t^2)(e^{2w_t} - e^{-2w_t}|\phi_t|^2). $$

Thus we deduce that

$$ (3.36) \qquad \ddot{w}_t + 4\dot{w}_t^2 \leq 0. $$

We have shown that as long as $(e^{2u_t}g, \phi_t)$ is almost-fuchsian, $\ddot{w}_t + 4\dot{w}_t^2 \leq 0$, as asserted.

Consider now the family of quadratic holomorphic differentials

$$ (3.37) \qquad \phi_{s,t} := s(1-t)\phi_0 + st\phi_1 $$

For all $s$, let $\mathcal{J}_s$ denote the set of all $t \in [0, 1]$ for which there exists a function $w_{s,t}$ making $(e^{2w_{s,t}}g, \phi_{s,t})$ into an almost Fuchsian pair. Let $\mathcal{S}$ denote the set of all $s \in [0, 1]$ for which $\mathcal{J}_s$ equals the interval $[0, 1]$. Trivially $\mathcal{S}$ contains 0. By Theorem 3.5, $\mathcal{S}$ is open. We now show that $\mathcal{S}$ is closed.

First, if $s_0 \in \bar{\mathcal{S}}$, the equicontinuity of all functions $(u_{s,t})$ for $s \in \mathcal{S}$, $t \in [0, 1]$ allows us to get a limiting family of functions which are solutions to Gauss-Codazzi equation. We claim that, for all $s$, the function $t \mapsto e^{-4w_{s,t}}|\phi_{s,t}|^2$ is convex over $\mathcal{J}_s$. Indeed,

$$ \begin{aligned} e^{4w_{s,t}} \partial_t^2 (e^{-4w_{s,t}} |\phi_{s,t}|^2) &= -4(\ddot{w}_{s,t} + 4\dot{w}_{s,t}^2) |\phi_{s,t}|^2 + 32\dot{w}_{s,t}^2 |\phi_{s,t}|^2 - 16\dot{w}_{s,t} g(\phi_{s,t}, \phi_{s,1} \\ & \qquad - \phi_{s,0}) + 2|\phi_{s,1} - \phi_{s,0}|^2. \\ &= -4(\ddot{w}_{s,t} + 4\dot{w}_{s,t}^2) |\phi_{s,t}|^2 + 2|4\dot{w}_{s,t}\phi_{s,t} - (\phi_{s,1} - \phi_{s,0})|^2. \end{aligned} $$

This is nonnegative, and convexity follows. By convexity, $e^{-4w_{s,t}}|\phi_{s,t}|^2$ is smaller than the maximal value for $t = 0$ and $t = 1$, and we deduce that it is smaller than $1 - \delta$, for $\delta > 0$ depending only on $\phi_0$ and $\phi_1$. It now readily follows that $\mathcal{S}$ is closed, so that, by connectedness, $1 \in \mathcal{S}$. In other words, $\mathcal{J}_1 = [0, 1]$, and the result follows. $\square$

We also have the following quantitative estimate for this convex set.

**Theorem 3.11.** The convex set $\mathfrak{I}(\mathcal{A}\mathcal{F})$ satisfies:

$$ (3.38) \qquad \{\phi : \sup_{z \in \mathcal{D}} |\phi(z)| < \frac{1}{2}\} \subset \mathfrak{I}(\mathcal{A}\mathcal{F}) \subset \{\phi : \sup_{z \in \mathcal{D}} |\phi(z)| < 1\} $$

*Proof.* Take $\phi$ with $|\phi|_{\infty} \leq \frac{1}{2}$. Then $-\frac{\ln 2}{2}$ and 0 are respectively sub and supersolutions of

$$ (3.39) \qquad \Delta u = -1 + e^{2u} + e^{-2u} |\phi|^2. $$

Hence by Proposition 2.4, we have a bounded solution satisfying Gauss-Codazzi, and necessarily $e^{-2u}|\phi| \leq 2|\phi|_{\infty} < 1$. Hence $\phi \in \mathfrak{I}(\mathcal{A}\mathcal{F})$. Conversely, if $\phi \in \mathfrak{I}(\mathcal{A}\mathcal{F})$, $u$ the solution to Gauss-Codazzi satisfies $u \leq 0$, and $|\phi|_{\infty} \leq |e^{-2u}\phi|_{\infty} < 1$. $\square$

Fixing now a subgroup $\Gamma$ of PSL$(2, \mathbb{R})$, one can look at the subset $\mathcal{A}\mathcal{F}_{\Gamma}$ of elements of $\mathcal{A}\mathcal{F}$ equivariant under some representation of $\Gamma$. The uniqueness properties shown imply that this set can be identified with a subset of $\mathrm{Hom}(\Gamma, \mathrm{PSL}(2, \mathbb{C}))/\mathrm{PSL}(2, \mathbb{C})$. In the case of a closed surface group, this subset are known as *almost-fuchsian representations* considered by Uhlenbeck [Uhl83].
Fix a hyperbolic surface $\Sigma_{g,n}$, and consider $\Gamma$ its fundamental group. Considering now holomorphic quadratic differentials equivariant under the action of $\Gamma$, identifying holomorphic quadratic differentials with the cotangent bundle to Teichmüller space of genus $(g, n)$, we get a smooth family of open bounded sets in each leaf. As these sets are isotropic (if $(e^{2u}g, \phi)$ is an almost-fuchsian disk, so is $(e^{2u}g, z\phi)$ for any $z$ of modulus one), it induces a Finsler metric on Teichmüller space, which is quasi-isometric to the Finsler metric dual to the $L^\infty$ unit balls in $T^*\mathcal{T}_g$.

**Corollary 3.12.** For $g \ge 2$ and $\mathcal{T}_g$ the Teichmüller space of genus $g$, there is a Finsler metric $p$ on $\mathcal{T}_g$ such that:

$$ (3.40) \qquad p(\phi) < 1 \iff \phi \in \mathfrak{S}(\mathcal{A}\mathcal{F}) $$

This Finsler metric is quasi-isometric to the Finsler metric dual to the $L^\infty$ unit balls in $T^*\mathcal{T}_g$.

*Remark.* It is interesting to compare this Finsler metric to the metrics dual to $L^p$-unit balls in $T^*\mathcal{T}_g$. For $p = 2$, one gets Weil-Petersson's metric, see Wolpert [Wol09]. For $p = 1$, one gets the Teichmüller metric, see Papadopoulos-Su [PS15].

*Remark.* With this discussion, we have a description of the fibration map from $\mathcal{A}\mathcal{F}_\Gamma \to \mathcal{T}_{g,n}$, which is simply the conformal class of the induced metric on $\Sigma_{g,n}$.
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