Non-convex balls in the Teichm\"uller metric

TL;DR

This paper demonstrates that in Teichmüller space for surfaces with genus g and p punctures, there exist non-convex balls in the Teichmüller metric when the complexity exceeds a certain threshold.

Contribution

It proves the existence of non-convex metric balls in Teichmüller space for surfaces with sufficient complexity, revealing new geometric properties.

Findings

Existence of non-convex balls in Teichmüller space for certain surfaces.

Non-convexity occurs when 3g - 3 + p > 1.

Provides geometric insight into the structure of Teichmüller space.

Abstract

We prove that the Teichm\"uller space of surfaces of genus $\mathbf{g}$ with $\mathbf{p}$ punctures contains balls which are not convex in the Teichm\"uller metric whenever $3\mathbf{g}-3+\mathbf{p} > 1$.