Stability of systolic inequalities for the Möbius strip and Klein bottle

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.