Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Teichmüller extremal maps on infinite Riemann surfaces

Dragomir Šarić

TL;DR

The paper provides a complete criterion for when a Teichmüller-type extremal map exists in the Teichmüller class of an infinite Riemann surface, via a height-map variational principle on the space of finite-area holomorphic quadratic differentials $A(X)$. It introduces the height map $f_{\#}:A(X)\to A(Y)$ and shows that the class $[f]\in T(X)$ has a Teichmüller-type extremal map iff the supremum $L=\sup_{\varphi\neq0}\max\{ \|f_{\#}(\varphi)\|_{L^1}/\|\varphi\|_{L^1}, \|\varphi\|_{L^1}/\|f_{\#}(\varphi)\|_{L^1} \}$ is attained at some $\varphi_{\max}$. When attained, the corresponding Teichmüller map stretches the horizontal foliation in the natural parameter of $\varphi_{\max}$ by the factor $L$. The paper develops a constructive approach: approximating $\varphi$ by Jenkins-Strebel differentials on exhaustions, proving convergence of the associated $f_{\#}$-images, and showing that attainment of $L$ forces a precise linear relation between $f_{\#}(\varphi_{\max})$ and $f_{\#}(-\varphi_{\max})$, which yields a Teichmüller map in the given homotopy class. A crucial continuity result ensures the approximations behave well under height variations, underpinning the existence proof and extending finite-type techniques to infinite-type surfaces. The work also highlights open problems, such as extending the criterion to general Fuchsian groups and characterizing Strebel/Busemann points through the height map.

Abstract

Let $X=\mathbb{D}/Γ$ be an arbitrary Riemann surface. We establish a necessary and sufficient criterion for $[f]\in T(X)$ to have a Teichmüller-type extremal map.

Teichmüller extremal maps on infinite Riemann surfaces

TL;DR

The paper provides a complete criterion for when a Teichmüller-type extremal map exists in the Teichmüller class of an infinite Riemann surface, via a height-map variational principle on the space of finite-area holomorphic quadratic differentials . It introduces the height map and shows that the class has a Teichmüller-type extremal map iff the supremum is attained at some . When attained, the corresponding Teichmüller map stretches the horizontal foliation in the natural parameter of by the factor . The paper develops a constructive approach: approximating by Jenkins-Strebel differentials on exhaustions, proving convergence of the associated -images, and showing that attainment of forces a precise linear relation between and , which yields a Teichmüller map in the given homotopy class. A crucial continuity result ensures the approximations behave well under height variations, underpinning the existence proof and extending finite-type techniques to infinite-type surfaces. The work also highlights open problems, such as extending the criterion to general Fuchsian groups and characterizing Strebel/Busemann points through the height map.

Abstract

Let be an arbitrary Riemann surface. We establish a necessary and sufficient criterion for to have a Teichmüller-type extremal map.

Paper Structure

This paper contains 7 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. A necessary condition for the existence of the Teichmüller extremal maps
  3. A sufficient condition for the existence of the Teichmüller extremal maps
  4. Approximation by differentials with single cylinders
  5. The heights of the negative of the maximum quadratic differential
  6. From maximum of the height function to the Teichmüller maps
  7. The continuity of the approximations with respect to the heights map

Key Result

Theorem 1.1

Let $X=\mathbb{D}/\Gamma$ be a Riemann surface with $\Gamma$ of the first kind. Then $[f]\in T(X)$ admits a Teichmüller-type extremal map if and only if the supremum is achieved at some $\varphi_{\mathrm{max}}\in A(X)\setminus\{ 0\}$. When the supremum is achieved, the Teichmüller map stretches the horizontal direction in the natural parameter of $\varphi_{\mathrm{max}}$ by the amount equal to th

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 2 more