On finite marked length spectral rigidity of hyperbolic cone surfaces and the Thurston metric

TL;DR

This paper proves that hyperbolic cone surfaces are uniquely determined by finitely many geodesic lengths and demonstrates that the Thurston metric extends to their Teichmüller space, nearly isometrically relating it to punctured hyperbolic surfaces.

Contribution

It establishes finite spectral rigidity for hyperbolic cone surfaces and extends the Thurston metric to their Teichmüller space, comparing it with classical hyperbolic surfaces.

Findings

Hyperbolic cone structures are determined by finitely many geodesic lengths.

The Thurston metric is well-defined on the Teichmüller space of cone surfaces.

The Teichmüller space of cone surfaces is almost isometric to that of punctured hyperbolic surfaces.

Abstract

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichm\"uller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichm\"uller space with the Teichm\"uller space of complete hyperbolic surfaces with punctures, by showing that the two spaces (endowed with the Thurston metric) are almost isometric.

Figures

Equations

$$\lambda(\Delta)=\left\{\begin{array}[]{ll}-\theta,&\text{ if }\Delta\text{ is a cone point of angle }\theta\in(0,\pi),\\ 0,&\text{ if }\Delta\text{ is a cusp},\\ l,&\text{ if }\Delta\text{ is a geodesic boundary of length }l>0.\end{array}\right.$$

$$F_{(\Gamma ,\mathcal{B})}:{\mathcal{T}}_{g,n}$$

$$(\Lambda ,L,T)$$

$$\{\begin{array}{l}|l_X([\alpha ])-l_{X^{\prime }}([\alpha ])|\le D{\sum }_{j=1}^{3g-3+n}i([\alpha ],[{\gamma }_j]),\\ \frac{1}{C}\le \frac{l_X([\alpha ])}{l_{X^{\prime }}([\alpha ])}\le C,\end{array}$$

$$d_{\mathrm{Th}}(X_1,X_2)=\log \underset{[\alpha ]\in \mathcal{S}(S)}{\sup }\frac{l_{X_2}([\alpha ])}{l_{X_1}([\alpha ])},$$

$$d_{\mathrm{Th}}(X_1^{\prime },X_2^{\prime })-C\le d_{\mathrm{Th}}(X_1,X_2)\le d_{\mathrm{Th}}(X_1^{\prime },X_2^{\prime })+C,forany{X}_1,X_2\in {\mathcal{T}}_{g,n},$$

$${\Psi }_{\Lambda }:{\mathcal{T}}_{g,n}(\Lambda )$$

$$X$$

$$\Pi :{\mathbb{R}}_{+}^{\mathcal{S}}$$

$$(s_{\alpha }{)}_{\alpha \in \mathcal{S}(S)}$$

Full text

AMS Mathematics Subject Classification (2010): 53C24, 51F99, 53B40.

This work is partially supported by NSFC, No: 11271378.

1. Introduction

Let $S=S_{g,n}$ be an oriented surface of genus $g$ with $n$ boundary components. We shall consider hyperbolic cone metrics on $S$, that is, hyperbolic structure on $S$ such that each boundary component is either a cone point (with cone angle strictly less than $\pi$), a puncture (i.e., a cone point with zero angle), or a simple closed geodesic with positive length. In this case, a boundary component $\Delta$ of $S$ is associated with a generalized length function (also called boundary assignment) $\lambda(\Delta)$, defined by

[TABLE]

1.1. Parametrization of Teichmüller space

Let $\mathcal{T}_{g,n}$ be the Teichmüller space of marked hyperbolic cone metrics on $S$. Given any $\Lambda=(\lambda_{1},...,\lambda_{n})\in(-\pi,\infty)^{n}$, we let $\mathcal{T}_{g,n}(\Lambda)$ be the subspace of $\mathcal{T}_{g,n}$ corresponding to hyperbolic cone surfaces whose boundary components have fixed generalized lengths $\Lambda$.

It is well known [FLP] that the Teichmüller space $\mathcal{T}_{g,n}(0)$ can be parameterized by the geodesic lengths of finitely many specified simple closed curves. Moreover, Seppälä and Sorvali [SS2] proved that only $6g-6+2n$ simple closed curves are required if $n\neq 0$. (For a closed surface of genus $g$, $6g-5$ simple closed curves are needed, see Schmutz [Sch].) A natural question is whether these results still hold for our enlarged Teichmüller space $\mathcal{T}_{g,n}$. In §LABEL:sec:mlss, we prove that:

**Theorem A **.

Let $S_{g,n}$ be a surface with $g\geq 1,n\geq 1$ or $g=0,n\geq 6$. The Teichmüller space $\mathcal{T}_{g,n}$ can be parameterized by the geodesic lengths of finitely many non-peripheral simple closed curves (see Definition 2) of $S_{g,n}$. Moreover, the minimal number of these parameters is less than $12g-12+32n$.

If $(g,n)\in\{(0,0),(0,1),(0,2),(0,3),(0,4),(0,5),(1,0)\}$, we call the surface $S_{g,n}$ exceptional. Otherwise, we call surface $S_{g,n}$ non-exceptional.

1.2. Comparisons of geometry between hyperbolic cone surfaces

Intuitively a cusp can be considered as a limit of cone points with angles tending to zero, or as a limit of geodesic boundaries with lengths tending to zero. We want to compare the geometries of hyperbolic cone surfaces under different assignments of cone angles and boundary lengths. In particular, we want to compare the geodesic lengths of simple closed curves on hyperbolic cone surfaces when we modify cone points or geodesic boundaries to cusps.

For a given pants decomposition $\Gamma=\{\gamma_{1},\cdots,\gamma_{3g-3+n}\}$ and a collection of seams $\mathcal{B}=\{\beta_{1},\cdots,\beta_{k}\}$, we define $F_{\Gamma,\mathcal{B}}$ by

[TABLE]

where $(\Lambda,L,T)$ and $(0,L,T)$ are the corresponding Fenchel-Nielsen (length-twist) coordinates (see Section LABEL:sec:fn). We denote the restriction of $F_{\Gamma,\mathcal{B}}$ on $\mathcal{T}_{g,n}(\Lambda)$ by $F_{\Gamma,\mathcal{B},\Lambda}$.

Theorem B **** (Length comparison inequalities).

Let $S$ be a non-exceptional surface, $\Gamma=\{\gamma_{1},\cdots,\gamma_{3g-3+n}\}$ be a pants decomposition and $\mathcal{B}=\{\beta_{1},\cdots,\beta_{k}\}$ be a collection of seams. There exist constants $C,D$ depending on $\Lambda$ such that for any $X\in\mathcal{T}_{g,n}(\Lambda)$, $X^{\prime}=F_{\Gamma,\mathcal{B},\Lambda}(X)$ and any isotopy class of non-peripheral simple closed curve $[\alpha]$,

[TABLE]

where $i(\cdot,\cdot)$ is the geometric intersection number, and ${l_{X}([\alpha])}$ is the length of the geodesic representative in $[\alpha]$. Moreover, $D\to 0,C\to 1$ as $\Lambda\to 0$.

1.3. The Thurston metric on $\mathcal{T}_{g,n}(\Lambda)$

There is an asymmetric metric on $\mathcal{T}_{g,n}(0)$ defined by

[TABLE]

which is the so called Thurston metric [Th].

We combine Theorem A with a generalized Mcshane’s identity on a hyperbolic cone surface due to Tam-Wong-Zhang [TWZ] to show that

**Theorem C **.

Let $S$ be a non-exceptional surface. For any $X_{1},\ X_{2}\in\mathcal{T}_{g,n}(\Lambda)$, if l_{X_{1}}([\alpha])\geq l_{X_{2}}([\alpha])\ for any $[\alpha]\in\mathcal{S}(S)$, then $X_{1}=X_{2}$.

As a result, we can define the Thurston metric on $\mathcal{T}_{g,n}(\Lambda)$ for any $\Lambda\in(-\pi,\infty)^{n}$, using the formula (2). It would be interesting to compare the Thurston metric with the arc metric on $\mathcal{T}_{g,n}(\Lambda)$, defined by Liu-Papadopoulos-Su-Théret [LPST1].

Using Theorem B, we are able to prove that any two Teichmüller spaces $\mathcal{T}_{g,n}(\Lambda),\mathcal{T}_{g,n}(\Lambda^{\prime})$ are almost isometric.

Theorem D **** (Almost-isometry).

Let $\Lambda\in(-\pi,\infty)^{n}$ and let $S$ be a non-exceptional surface. The map $F_{\Gamma,\mathcal{B},\Lambda}:\mathcal{T}_{g,n}(\Lambda)\to\mathcal{T}_{g,n}(0)$ is an almost-isometry, i.e., there is a constant $C$ depending on $\Lambda$ such that

[TABLE]

where $X^{\prime}_{1}=F_{\Gamma}(X_{1})$ and $X^{\prime}_{2}=F_{\Gamma}(X_{2})$. Moreover, $C\to 0$ as $\Lambda\to 0$.

1.4. The Thurston boundary of $\mathcal{T}_{g,n}(\Lambda)$

For the space $\mathcal{T}_{g,n}(0)$, the Thurston boundary is naturally identified with $\mathcal{PML}(S)$, the space of projective classes of measured laminations ([FLP]). We will prove that this is also true in our settings. Denote by $\mathbb{R}_{+}^{\mathcal{S}}$ the set of non-negative functionals on the set of isotopy classes of non-peripheral simple closed curves (see Definition 2).

Theorem E**.**

Suppose $\Lambda\in(-\pi,\infty)^{n}$ and $S$ is a non-exceptional surface. Let $\Psi_{\Lambda}$ and $\Pi$ be the maps defined as following:

[TABLE]

and

[TABLE]

Then

1. (a)

both $\Psi_{\Lambda}$ and $\Pi\circ\Psi_{\Lambda}$ are embeddings, where $\mathcal{T}_{g,n}(\Lambda)$ is equipped with the topology induced by $d_{\mathrm{Th}}$ and $\mathbb{R}^{\mathcal{S}}_{+}$ is equipped with the weak topology; 2. (b)

$\mathcal{T}_{g,n}(\Lambda)\ni X_{n}\to\xi\in\mathcal{PML}(S)\iff\mathcal{T}_{g,n}(0)\ni F_{\Gamma,\mathcal{B},\Lambda}(X_{n})\to\xi\in\mathcal{PML}(S)$. As a result, the boundary of $\mathcal{T}_{g,n}(\Lambda)$ in $P\mathbb{R}^{\mathcal{S}}_{+}$ (i.e., the Thurston boundary of $\mathcal{T}_{g,n}(\Lambda)$) is homeomorphic to $\mathcal{PML}(S)$.

The paper is organized as following. In Section 2, we recall basic facts about hyperbolic cone surfaces. In Section LABEL:sec:fn, we study the Fenchel-Nielsen coordinates of the Teichmüller space $\mathcal{T}_{g,n}(S)$. In Section LABEL:sec:rigidity, we prove Theorem A and Theorem C. In Section LABEL:sec:comp-Y, we compare the lengths of simple closed curves between hyperbolic cone surfaces and prove Theorem B. In Section LABEL:sec:al-iso, we prove Theorem D. In Section LABEL:sec:thurstonb, we study some basic properties of the Teichmüller space $\mathcal{T}_{g,n}(\Lambda)$ and prove Theorem E.

**Acknowledgements ** I would like to thank Lixin Liu and Weixu Su for their useful suggestions for improving this manuscript. I also thank the referee for numerous comments and suggestions.

2. Hyperbolic cone metrics

In this section, we study some basic properties of hyperbolic cone surfaces, with or without boundary, and with each cone angle less than $\pi$.

Definition 1** ([CHK]).**

A hyperbolic cone-surface is a two-dimensional manifold $X$, with or without boundary, which can be triangulated by hyperbolic triangles.

The singular locus $Cone(X)$ of a hyperbolic cone-surface $X$ consists of interior point of $X$ which have no neighbourhoods isometric to a ball in the hyperbolic plane and boundary points which have no neighbourhoods isometric to a half ball in the hyperbolic plane. It follows that

• •

$Cone(X)$ is contained in the set of vertices of the hyperbolic triangulation of $X$, and it is a finite set;

• •

At each point of $Cone(X)$, there is a cone angle which is the sum of the angles of the dihedral angles of the triangles containing the point;

• •

$X\backslash Cone(X)$ has a smooth Riemannian metric of constant curvature $-1$, but this metric is incomplete if there is a cone point with positive cone angle;

In this paper, we are interested in the case where $X$ satisfies the following requirements:

1. (a)

X has at most $n$ cone points and every cone angle is strictly smaller than $\pi$. In particular, a cone point with zero cone angle is called a cusp. 2. (b)

Every boundary component of $X$ consists of at most one geodesic.

Remark 1**.**

To distinguish a positive cone angle point and the zero angle point, whenever we mention a cone point we mean a cone point with a positive cone angle.

Definition 2**.**

A generalized boundary component of a cone surface $X$ is a geodesic boundary component, a cusp, or a cone point. We denote by $\Sigma_{X}$ the set of all generalized boundary components of $X$. A non-trival simple closed curve on $X$ is called non-peripheral if it is not homotopic to any generalized boundary component. A simple arc with endpoints on the generalized boundary components is called non-peripheral if it is not homotopic to any subarc of the generalized boundary components.

Definition 3**.**

A marked hyperbolic cone surface is a pair $(f,X)$, where $X$ is a hyperbolic cone-surface, and $f:S\to X$ is a homeomorphism. Two marked hyperbolic cone-surfaces $(f,X)$ and $(f^{\prime},X^{\prime})$ are called equivalent if there is an isometry isotopic to $f\circ(f^{\prime})^{-1}$.

A necessary condition of two marked hyperbolic cone surfaces to be equivalent is that they have the same numbers of cone points, of cusps and of geodesic boundaries. Denote by $\mathcal{T}_{g,n}(S)$ the space of equivalent classes of marked hyperbolic cone surfaces.