A Teichmuller space for negatively curved surfaces

TL;DR

This paper introduces a novel Teichmuller space for negatively curved surfaces by extending classical concepts to Finsler metrics and using Higgs bundle techniques, providing a new framework for understanding surface geometry.

Contribution

It develops a new Teichmuller space for negatively curved surfaces using Finsler metrics and Higgs bundle methods, generalizing classical Teichmuller theory.

Findings

Defines a flat connection with Hamiltonian diffeomorphism holonomy.

Constructs a moduli space extending classical Teichmuller space.

Provides an alternative approach to surface geometry via CR functions.

Abstract

We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of S^1 x R. Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite dimensional group and focuses on a family of metrics, relying on a construction of O.Biquard, which is parametrized by the infinite-dimensional space of CR functions on the unit circle bundle of a hyperbolic surface. This generates an alternative approach to defining a connection and offers the possibility of this vector space representing a moduli space which generalizes and includes the classical Teichmueller space.