# On domains biholomorphic to Teichm\"{u}ller spaces

## TL;DR

This paper proves that Teichmüller spaces of closed surfaces of genus at least 2 cannot be biholomorphically equivalent to any domain that exhibits local strict convexity at some boundary point, revealing geometric constraints.

## Contribution

It establishes a new geometric restriction on Teichmüller spaces, showing they are not biholomorphic to locally strictly convex domains at boundary points.

## Key findings

- Teichmüller spaces are not biholomorphic to locally strictly convex domains.
- The result constrains the types of domains biholomorphic to Teichmüller spaces.
- Provides insight into the complex geometry of Teichmüller spaces.

## Abstract

We prove that the Teichm\"{u}ller space $\mathscr{T}$ of a closed surface of genus $g \ge 2$ cannot be biholomorphic to any domain which is locally strictly convex at some boundary point.