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Totally geodesic submanifolds of Teichmuller space

Alex Wright

Abstract

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.

Totally geodesic submanifolds of Teichmuller space

Abstract

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.

Paper Structure

This paper contains 3 sections, 18 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['T:finite']} and \ref{['T:trans']}
  3. Proof of Theorem \ref{['T:main']}

Key Result

Theorem 1.1

The image in $\mathcal{M}_{g,n}$ of a totally geodesic complex submanifold of $\mathcal{T}_{g,n}$ of dimension greater than 1 is a closed totally geodesic subvariety of $\mathcal{M}_{g,n}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['T:trans']}
  • proof : Proof of Theorem \ref{['T:finite']} given Theorem \ref{['T:trans']}.
  • Lemma 3.1
  • ...and 24 more