Almost isometries between Teichmüller spaces
We prove that the Teichmüller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmüller metric) is almost isometric to the Teichmüller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmüller metric).
Keywords: Teichmüller space, almost isometry, Thurston metric, Teichmüller metric, arc metric.
AMS MSC2010: 32G15, 30F60, 51F99.
Let be an oriented surface of genus with boundary components such that . The Euler characteristic of is . Throughout this paper we assume that . Recall that a marked complex structure on is a pair where is a Riemann surface and is an orientation preserving homeomorphism. Two marked complex structures and are called equivalent if there is a conformal map homotopic to . Denote by the equivalence class of . The set of equivalence classes of marked complex structures is the Teichmüller space denoted by .
Let be a Riemann surface with boundary. There exist two different hyperbolic metrics on . One is of infinite area obtained from the Uniformization theorem, the other one is of finite area obtained from the restriction to 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 . 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 is a hyperbolic surface equipped with an orientation-preserving homeomorphism , where maps each component of the boundary of to a geodesic boundary of . Two marked hyperbolic surfaces and are called equivalent if there is an isometry homotopic to relative to the boundary. The Teichmüller space is also the set of equivalence classes of marked hyperbolic surface. For simplicity, we will denote a point in by , without explicit reference to the marking or to the equivalence relation.
Let be the boundary components of . For any . Let be the set of the equivalence classes of marked hyperbolic metrics whose boundary components have hyperbolic lengths . In particular, is the Teichmüller space of surfaces with punctures. It is clear that . Let be a pants decomposition of , i.e. the complement of on consists of pairs of pants . Let be a set of disjoint simple closed curves whose restriction to any pair of pants consists of three arcs, such that any two of the arcs are not free homotopic with respect to the boundary of . The pair is called a marking of . For any , let be the corresponding Fenchel-Nielsen coordinates with respect to the marking , where represents the lengths of , represents the twists along and 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 and in the following way:
The goal of this paper is to compare various metrics on the Teichmüller spaces and via the homeomorphism .
Two metric spaces and are called almost isometric if there exist a map , two positive constants and , such that both of the following two conditions hold.
For any ,
For any , there exists such that
1.1. The Thuston metric and the arc metric
An essential simple closed curve on 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 be the set of homotopy classes of essential simple closed curves on S, be the set of homotopy classes of essential arcs on S, and be the set of homotopy classes of the boundary components.
For any , define
From the works [Pan] and [LPST2], both and are asymmetric metric on , which are called the Thurston metric and the arc metric respectively. Moreover, the authors ([LPST2]) observed that
Our first result is the following.
and are almost isometric. More precisely, there is a constant depending on the surface and boundary lengths such that,
Papadopoulos-Su ([PS]) considered the case where is close to zero, they showed that the constant in Theorem 1.2 will tend to zero if tends to zero.
Proof of Theorem 1.2.
The arc metric and the Thurston metric are almost-isometric in . More precisely, there is a constant depending on the surfaces and boundary lengths such that,