On sets of marked once-holed tori allowing holomorphic mappings into Riemann surfaces with marked handle

TL;DR

This paper studies the geometric structure of the set of marked once-holed tori that admit holomorphic mappings into a fixed Riemann surface with a marked handle, revealing boundary properties and extremal length criteria.

Contribution

It demonstrates that the boundary of this set is always non-smooth and provides a way to evaluate the critical extremal length for holomorphic mappings.

Findings

Boundary of the set is never smooth

Set is a closed domain with Lipschitz boundary

Critical extremal length is expressed via hyperbolic lengths

Abstract

In our previous work, for a given Riemann surface $Y_0$ with marked handle, we investigated geometric properties of the set of marked once-holed tori $X$ allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.