Weil-Petersson Teichm\"uller theory of surfaces of infinite conformal type

TL;DR

This paper surveys the extension of Weil-Petersson Teichm"uller theory to infinite-dimensional spaces, covering geometric structures, analysis, and applications in physics and other fields.

Contribution

It provides a comprehensive overview of the Weil-Petersson geometry of infinite-type Teichm"uller spaces, including definitions, properties, and interdisciplinary applications.

Findings

Rigorous definitions of complex Hilbert manifold structures

Development of K"ahler geometry and global analysis in infinite dimensions

Connections to physics, fluid mechanics, and conformal field theory

Abstract

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichm\"uller spaces of infinite type, including for example the universal Teichm\"uller space. In this paper we give a survey of the main results in the Weil-Petersson geometry of infinite-dimensional Teichm\"uller spaces. This includes the rigorous definition of complex Hilbert manifold structures, K\"ahler geometry and global analysis, and generalizations of the period mapping. We also discuss the motivations of the theory in representation theory and physics beginning in the 1980s. Some examples of the appearance of Weil-Petersson Teichm\"uller space in other fields such as fluid mechanics and two-dimensional conformal field theory are also provided.