Stability of systolic inequalities for the M\"obius strip and Klein bottle

TL;DR

This paper provides an alternative proof of Bavard's systolic inequalities for the Klein bottle and M"obius strip, including stability estimates relating the systolic defect to conformal factor deviations.

Contribution

It offers new proofs and stability estimates for systolic inequalities on the Klein bottle and M"obius strip, extending Bavard's results with quantitative bounds.

Findings

Established an estimate on the systolic defect in terms of conformal factor distance.

Provided an alternative proof of Bavard's systolic inequalities.

Extended stability results to the M"obius strip.

Abstract

The systolic area $\alpha_{sys}$ of a nonsimply connected compact Riemannian surface $(M,g)$ is defined as its area divided by the square of the systole, where the systole is equal to the length of a shortest noncontractible closed curve. The systolic inequality due to Bavard states that on the Klein bottle, the systolic area has the optimal lower bound $\frac{2\sqrt{2}}{\pi}$. Bavard also constructed metrics of minimal systolic area in any given conformal class. We give an alternative proof of these results, which also yields an estimate on the systolic defect $\alpha_{sys}-\frac{2\sqrt{2}}{\pi}$ in terms of the $L^2$-distance of the conformal factor to the metric which minimizes the systolic area. On the M\"obius strip, we also prove similar estimates for metrics in fixed conformal classes.