Year: 2025
Type: article
Abstract: The systolic area $$\alpha _{sys}$$ 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 $$\frac{2\sqrt{2}}{\pi }$$ 2 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 $$\alpha _{sys}-\frac{2\sqrt{2}}{\pi }$$ α sys - 2 2 π in terms of the $$L^2$$ 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.
Source: Journal of Geometric Analysis
Author: Jan Eyll
Institution: Ruhr University Bochum
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: Ruhr-Universität Bochum
OpenAlex ID: W4414409636