New degeneration phenomenon for infinite-type Riemann surfaces

TL;DR

This paper constructs explicit examples demonstrating a new degeneration phenomenon in the Bers boundary of infinite-type Riemann surfaces, revealing complex boundary behaviors in their Teichmüller spaces.

Contribution

It provides the first concrete examples of degenerations in the Bers boundary for infinite-type Riemann surfaces, expanding understanding of their boundary structures.

Findings

Existence of Riemann surfaces in the Bers boundary with specific homeomorphisms.

Many such degenerations are shown to exist.

The results highlight complex boundary phenomena for infinite-type surfaces.

Abstract

Since the Teichm\"uller space of a surface $R$ is a deformation space of complex structures defined on $R$, its Bers boundary describes the degeneration of complex structures in a certain sense. In this paper, constructing a concrete example, we prove that if S is a Riemann surface of infinite type, there exists a Riemann surface with the marking, which is homeomorphic to the surface $R$ in the Bers boundary. We also show that many such degenerations exist in the Bers boundary.