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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.13715 |
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Table of Contents:
- The systolic area $α_{sys}$ 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 $\frac{2\sqrt{2}}π$. 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 $α_{sys}-\frac{2\sqrt{2}}π$ 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.