Maximal cusps are not dense

TL;DR

This paper demonstrates that maximal cusps are not dense on the Bers boundary of certain infinite-type Riemann surfaces, providing a counterexample to a known finite-type case result and exploring the structure of related subsets.

Contribution

It establishes that maximal cusps do not densely approach Bers boundary points in infinite-type surfaces and shows these subsets form a manifold, extending understanding beyond finite-type cases.

Findings

Maximal cusps are not dense on the Bers boundary for certain infinite-type surfaces.

The subset of Teichmüller space with these properties has a manifold structure.

Counterexample to McMullen's finite-type result in the infinite-type setting.

Abstract

We proved that the Maximal cusp is not dense on the Bers boundary of the Teichm\"uller space of infinite type Riemann surfaces satisfying some analytic conditions. This is a counterexample to the infinite-type case of the McMullen result for finite-type Riemann surfaces. More precisely, we showed that maximal cusps cannot approach the points on the Bers boundary corresponding to the deformation by the David map, which can be regarded as a degenerate quasiconformal map in the neighborhood of one end. In addition, to prove this, we used quasiconformal deformations in the neighborhood of a fixed end. We then proved that such a subset of the Teichm\"uller space has a manifold structure.