Almost isometries between Teichm\"uller spaces

TL;DR

This paper demonstrates that Teichm"uller spaces of surfaces with boundary, when equipped with certain metrics, are nearly isometric to spaces of punctured surfaces with corresponding metrics, revealing deep geometric connections.

Contribution

It establishes almost isometries between different Teichm"uller spaces with boundary and punctured surfaces under specific metrics, providing new insights into their geometric relationships.

Findings

Teichm"uller space with boundary and arc metric is almost isometric to punctured surface space with Thurston metric.

Teichm"uller space with boundary and Teichm"uller metric is almost isometric to punctured surface space with the same metric.

The results reveal a near-isometric correspondence between these moduli spaces under natural metrics.

Abstract

We prove that the Teichm\"uller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichm\"uller metric) is almost isometric to the Teichm\"uller space of punctured surfaces equipped with the Thurston metric (resp. the Teichm\"uller metric).

Figures

Equations

$${\Phi }_{\Gamma }:T_{g,n}(\Lambda )$$

$$(\mathcal{L},\mathcal{T},\Lambda )$$

$$|d_2(f(x),f(y))-d_1(x,y)|\le B.$$

$$d_2(z,f(x))\le A.$$

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

$$d_A(X_1,X_2):=\log \underset{[\alpha ]\in \mathcal{A}(S)}{\sup }\frac{l_{X_2}([\alpha ])}{l_{X_1}([\alpha ])}.$$

$$d_A(X_1,X_2)=\log \underset{[\alpha ]\in \mathcal{A}(S)\cup \mathcal{B}(S)\cup \mathcal{S}(S)}{\sup }\frac{l_{X_2}([\alpha ])}{l_{X_1}([\alpha ])}.$$

$$|d_A(X_1,X_2)-d_{Th}({\Phi }_{\Gamma }(X_1),{\Phi }_{\Gamma }(X_2))|\le C_1.$$

$$|d_A(X_1,X_2)-d_{Th}(X_1,X_2)|\le C_2.$$

Full text

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

Keywords: Teichmüller space, almost isometry, Thurston metric, Teichmüller metric, arc metric.

AMS MSC2010: 32G15, 30F60, 51F99.

1. introduction

Let $S$ be an oriented surface of genus $g$ with $n$ boundary components such that $n\geq 1$. The Euler characteristic of $S$ is $\chi(S)=2-2g-n$. Throughout this paper we assume that $\chi(S)\leq 0$. Recall that a marked complex structure on $S$ is a pair $(X,f)$ where $X$ is a Riemann surface and $f:S\to X$ is an orientation preserving homeomorphism. Two marked complex structures $(X,f)$ and $(Y,h)$ are called equivalent if there is a conformal map homotopic to $f\circ h^{-1}$. Denote by $[X,f]$ the equivalence class of $(X,f)$. The set of equivalence classes of marked complex structures is the Teichmüller space denoted by $T_{g,n}$.

Let $X$ be a Riemann surface with boundary. There exist two different hyperbolic metrics on $X$. One is of infinite area obtained from the Uniformization theorem, the other one is of finite area obtained from the restriction to $X$ of the hyperbolic metric on its (Sckottky) double such that each boundary component is a smooth simple closed geodesic (see §LABEL:ssec:double). The second one is called the intrinsic metric on $X$. In this paper when we mention a hyperbolic metric on a surface with nonempty boundary we mean the second one. The correspondence between complex structure and hyperbolic metric provides another approach for the Teichmüller theory. Recall that a * marked hyperbolic surface* $(X,f)$ is a hyperbolic surface $X$ equipped with an orientation-preserving homeomorphism $f:S\rightarrow X$, where $f$ maps each component of the boundary of $S$ to a geodesic boundary of $X$. Two marked hyperbolic surfaces $(X,f)$ and $(Y,h)$ are called equivalent if there is an isometry homotopic to $f\circ{h}^{-1}$ relative to the boundary. The Teichmüller space $T_{g,n}$ is also the set of equivalence classes of marked hyperbolic surface. For simplicity, we will denote a point $[X,f]$ in ${T}_{g,n}$ by $X$, without explicit reference to the marking or to the equivalence relation.

Let $\beta_{1},\cdots,\beta_{n}$ be the boundary components of $S$. For any $\Lambda=(\lambda_{1},...,\lambda_{n})$ $\in R_{\geq 0}^{n}$. Let $T_{g,n}(\Lambda)\subset T_{g,n}$ be the set of the equivalence classes of marked hyperbolic metrics whose boundary components have hyperbolic lengths $(l(\beta_{1}),...,l(\beta_{n}))=\Lambda$. In particular, $T_{g,n}(0)$ is the Teichmüller space of surfaces with $n$ punctures. It is clear that $T_{g,n}=\cup_{\Lambda\in\mathbb{R}^{n}_{+}}T_{g,n}(\Lambda)$. Let $\Gamma=\{\gamma_{1},\cdots,\gamma_{3g-3+n}\}$ be a pants decomposition of $S$, i.e. the complement of $\Gamma$ on $S$ consists of $2g-2+n$ pairs of pants $\{R_{i}\}_{i=1}^{2g-2+n}$. Let $\mu$ be a set of disjoint simple closed curves whose restriction to any pair of pants $R_{i}$ consists of three arcs, such that any two of the arcs are not free homotopic with respect to the boundary of $R_{i}$. The pair $(\Gamma,\mu)$ is called a marking of $S$. For any $X\in T_{g,n}$, let $(\mathcal{L},\mathcal{T},\Lambda)$ be the corresponding Fenchel-Nielsen coordinates with respect to the marking $(\Gamma,\mu)$, where $\mathcal{L}=(l_{1},\cdots,l_{3g-3+n})$ represents the lengths of $\{\gamma_{1},\cdots,\gamma_{3g-3+n}\}$, $\mathcal{T}=(t_{1},\cdots,t_{3g-3+n})$ represents the twists along $\{\gamma_{1},\cdots,\gamma_{3g-3+n}\}$ and $\Lambda=(\lambda_{1},...,\lambda_{n})$ represents the lengths of the boundary components (for details about Fenchel-Nielsen coordinates we refer to [Bu]). The Fenchel-Nielsen coordinates induce a natural homeomorphism between Teichmüller spaces $T_{g,n}(\Lambda)$ and $T_{g,n}(0)$ in the following way:

[TABLE]

The goal of this paper is to compare various metrics on the Teichmüller spaces $T_{g,n}(\Lambda)$ and $T_{g,n}(0)$ via the homeomorphism $\Phi_{\Gamma}$.

Definition 1.1**.**

Two metric spaces $(X_{1},d_{1})$ and $(X_{2},d_{2})$ are called almost isometric if there exist a map $f:X_{1}\to X_{2}$, two positive constants $A$ and $B$, such that both of the following two conditions hold.

1. (1)

For any $x,y\in X_{1}$,

[TABLE] 2. (2)

For any $z\in X_{2}$, there exists $x\in X_{1}$ such that

[TABLE]

1.1. The Thuston metric and the arc metric

An essential simple closed curve on $S$ is a simple closed curve which is not homotopic to a single point or a boundary component. An essential arc is a simple arc whose endpoints are on the boundary and which is not homotopic to any subarc of the boundary. Let ${\mathcal{S}(S)}$ be the set of homotopy classes of essential simple closed curves on S, ${\mathcal{A}(S)}$ be the set of homotopy classes of essential arcs on S, and $\mathcal{B}(S)$ be the set of homotopy classes of the boundary components.

For any $X_{1},X_{2}\in T_{g,n}(\Lambda)$, define

[TABLE]

and

[TABLE]

From the works [Pan] and [LPST2], both $d_{Th}$ and $d_{A}$ are asymmetric metric on $T_{g,n}(\Lambda)$, which are called the Thurston metric and the arc metric respectively. Moreover, the authors ([LPST2]) observed that

[TABLE]

Our first result is the following.

Theorem 1.2**.**

$(T_{g,n}(\Lambda),d_{A})$* and $(T_{g,n}(0),d_{Th})$ are almost isometric. More precisely, there is a constant $C_{1}$ depending on the surface $S$ and boundary lengths $\Lambda$ such that,*

[TABLE]

Remark 1**.**

Papadopoulos-Su ([PS]) considered the case where $\Lambda$ is close to zero, they showed that the constant $C_{1}$ in Theorem 1.2 will tend to zero if $\Lambda$ tends to zero.

Proof of Theorem 1.2.

To prove Theorem 1.2, it suffices to verify that they satisfy the two conditions in Definition 1.1. The first condition follows from Theorem 1.3 and Theorem LABEL:thm:thu-almost. The second condition follows from the fact that $\Phi_{\Gamma}$ is a homeomorphism. ∎

Theorem 1.3**.**

The arc metric and the Thurston metric are almost-isometric in $T_{g,n}(\Lambda)$. More precisely, there is a constant $C_{2}$ depending on the surfaces $S$ and boundary lengths $\Lambda$ such that,

[TABLE]