Sharp systolic inequalities for rotationally symmetric 2-orbifolds

TL;DR

This paper establishes upper bounds for systolic ratios on rotationally symmetric spindle orbifolds, showing these bounds are achieved by Besse metrics where all geodesics are closed.

Contribution

It introduces sharp systolic inequalities for a class of symmetric orbifolds and characterizes the extremal metrics as Besse metrics with closed geodesics.

Findings

Systolic ratios are globally bounded on rotationally symmetric spindle orbifolds.

The maximum systolic ratio is attained by Besse metrics.

All geodesics are closed in the extremal Besse metrics.

Abstract

We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.