Harmonic maps and wild Teichm\"uller spaces

TL;DR

This paper extends the parametrization of Teichmüller spaces to crowned hyperbolic surfaces using meromorphic quadratic differentials with higher order poles, involving harmonic maps with prescribed principal parts.

Contribution

It introduces a new parametrization of Teichmüller spaces for crowned hyperbolic surfaces via meromorphic quadratic differentials, generalizing Wolf's work for closed surfaces.

Findings

Existence of harmonic maps with prescribed principal parts.

Parametrization of Teichmüller space using meromorphic quadratic differentials.

Extension of Wolf's parametrization to non-compact surfaces.

Abstract

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichm\"uller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf's parametrization of the Teichm\"uller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.