The complex geometry of Teichm\"uller spaces and bounded symmetric domains

TL;DR

This paper explores the complex geometry of Teichmüller spaces and bounded symmetric domains, focusing on isometric maps in their Kobayashi metrics, highlighting similarities, differences, and open questions in their structure.

Contribution

The paper reviews recent results on isometric maps between Teichmüller spaces and bounded symmetric domains, emphasizing their complex analytic properties and open problems.

Findings

Identifies key features shared by the spaces

Highlights differences in their geometric structures

Presents open questions for future research

Abstract

We study isometric maps between Teichm\"uller spaces and bounded symmetric domains in their intrinsic Kobayashi metric. From a complex analytic perspective, these two important classes of geometric spaces have several features in common but also exhibit many differences. The focus here is on recent results proved by the author; we give a list of open questions at the end. Paper based on plenary talk presented to the 6th Ahlfors-Bers Colloquium at Yale, 23-26 October 2014.