Polygons and non-starlikeness of Teichmuller spaces

TL;DR

This paper proves that Teichmuller spaces of closed Riemann surfaces of genus g ≥ 2 are not starlike in Bers' embedding, using geometric properties of polygons, completing the understanding of their geometric structure.

Contribution

The paper provides a complete proof that Teichmuller spaces of closed genus g ≥ 2 are non-starlike, extending previous results and employing polygon-based geometric methods.

Findings

Teichmuller spaces of genus g ≥ 2 are not starlike in Bers' embedding.

The proof uses geometric features of polygons to establish non-starlikeness.

Completes the classification of geometric properties of these Teichmuller spaces.

Abstract

The problem of starlikeness of Teichmuller spaces in Bers' embedding was raised in 1974 and is solved (negatively) for Teichmuller spaces of sufficiently large dimensions. The original proof given by the author relies on the existence of conformally rigid domains established by Thurston. Later the author found another proof of non-starlikeness of universal Teichmuller space based on geometric features of rectilinear polygons. This paper provides a complete solution of the problem for Teichmuller spaces $T(g,0)$ of closed Riemann surfaces of genus $g \ge 2$.