Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Caratheodory metrics on Teichmuller spaces

Yiran Lin, Vladimir Markovic

Abstract

Let $S$ be an arbitrary Riemann surface whose Teichmüller space $T(S)$ has dimension at least two. A long standing problem is to determine whether the Carathéodory metric $d_C$ agrees with the Teichmüller metric $d_T$ on $T(S)$. It was shown that $d_C\ne d_T$ when $S$ is a closed surface of genus at least two. In this paper we study the general case, and prove that $d_C\ne d_T$ on $T(S)$ except possibly on the following seven Teichmüller spaces: $T^1_{0,0}$, $T^1_{0,1}$, $T^2_{0,0}$, $T^1_{0,2}$, $T^2_{0,1}$, $T^3_{0,0}$, and $T^3_{0,1}$.

Caratheodory metrics on Teichmuller spaces

Abstract

Let be an arbitrary Riemann surface whose Teichmüller space has dimension at least two. A long standing problem is to determine whether the Carathéodory metric agrees with the Teichmüller metric on . It was shown that when is a closed surface of genus at least two. In this paper we study the general case, and prove that on except possibly on the following seven Teichmüller spaces: , , , , , , and .
Paper Structure (51 sections, 42 theorems, 123 equations, 2 figures)

This paper contains 51 sections, 42 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $\Sigma_{g,n}$-accommodating surfaces
  3. Model Teichmüller spaces
  4. Totally ramified coverings and a 3-punctured disc
  5. Outline
  6. Organization
  7. The Carathéodory and Kobayashi pseudometrics
  8. Holomorphic projections
  9. Proof of Theorem \ref{['theorem: main theorem']}
  10. Proof of Lemma \ref{['lemma-4']}
  11. Hyperbolic geometry of quasiconformal maps
  12. Injectivity radius and quasiconformal homeomorphisms
  13. Think-Thin decompositions
  14. Essential conformal embeddings
  15. Teichmüller spaces and Beltrami differentials
  16. ...and 36 more sections

Key Result

Theorem 1.1

Let $S$ be a Riemann surface such that $\dim(\mathcal{T}(S))\ge 2$. Then $\mathbf{d}_C\neq \mathbf{d}_\mathcal{T}$ on $\mathcal{T}(S)$ unless $\mathcal{T}(S)$ is one from the following list: $\mathcal{T}^1_{0,0}$, $\mathcal{T}^1_{0,1}$, $\mathcal{T}^2_{0,0}$, $\mathcal{T}^1_{0,2}$, $\mathcal{T}^2_{0

Figures (2)

  • Figure 1: $L$-shaped polygon $L(a,b,q)$
  • Figure 2: $L(a_0,0,q_0-t)$

Theorems & Definitions (107)

  • Remark 1
  • Remark 2
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof
  • Theorem 1.5
  • Lemma 1.6
  • proof
  • ...and 97 more