Relative systoles in hyperelliptic translation surfaces

TL;DR

This paper investigates the behavior of the systole function on translation surfaces, showing that local maxima that are not global occur only in non-hyperelliptic components, revealing geometric distinctions.

Contribution

It establishes a precise criterion linking hyperelliptic structures to the nature of local maxima of the systole function on translation surfaces.

Findings

Local maxima of the systole function exist only in non-hyperelliptic components.

Hyperelliptic components do not admit non-global local maxima of the systole.

The result characterizes the geometric structure of translation surfaces based on systole behavior.

Abstract

In this paper we prove that the systole fonction on a connected component of area one translation surfaces admits a local maximum that is not a global maximum if and only if the connected component is not hyperelliptic.