Hyperellipticity and Systoles of Klein Surfaces

TL;DR

This paper develops a method to analyze hyperelliptic Klein surfaces by constructing companion Klein bottles, improving systolic inequalities through length bounds and real structure analysis, with special focus on Dyck's surface.

Contribution

It introduces a novel approach to study systoles on Klein surfaces via companion Klein bottles and extends existing inequalities using geometric and topological techniques.

Findings

Constructed companion Klein bottles for hyperelliptic Klein surfaces.

Improved systolic inequality bounds for Klein surfaces.

Analyzed length bounds of loops on companion surfaces and their relation to original surfaces.

Abstract

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard's short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov's. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original nonorientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the real locus. Of particular interest is the case of Dyck's surface. We also exploit an optimal systolic bound for the M\"obius band, due to Blatter.