Harmonic maps and wild Teichmüller spaces

2017/08/16 by Gupta, Subhojoy · 1 citation
#30F30 #30F60 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1708.04780

Abstract

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller 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 Teichmüller 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.

Cited by

• Harmonic maps and framed PSL₂(ℂ)-representations

Related

• The harmonic maps compactification of Teichmüller spaces for punctured Riemann surfaces
• Limits of harmonic maps and crowned hyperbolic surfaces
• Harmonic maps of punctured surfaces to the hyperbolic plane
• Holomorphic quadratic differentials in Teichmüller theory
• Uniform degeneration of hyperbolic surfaces with boundary along harmonic map rays