Systoles in translation surfaces

TL;DR

This paper investigates the maximum values of the shortest saddle connection length on translation surfaces, characterizing local maxima and exploring their relation to the number of shortest saddle connections.

Contribution

It provides a characterization of systole maxima on strata and constructs examples of local but not global maxima for genus at least three.

Findings

Characterization of maxima of the systole function.

Construction of examples with local maxima.

Analysis of the relation between systole maxima and shortest saddle connections.

Abstract

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation between (locally) maximal values of the systole function and the number of shortest saddle connections.