Stability of systolic inequalities for the Möbius strip and Klein bottle

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 Möbius strip, we also prove similar estimates for metrics in fixed conformal classes.

Minor changes, added References, removed proof of Lemma 2.1 as it can be found in the literature. Replaced with final version, published in the Journal of Geometric Analysis

References in corpus (1)

• A systolic inequality with remainder in the real projective plane

Cited by in corpus (1)

• Klein bottle and optimal systolic inequality for nonpositively curved surfaces