通过填充极小性得到度量曲面的收缩不等式
Systolic inequalities for metric surfaces via filling minimality
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中文总结 AI 辅助
研究与亏格为1的环面或实射影平面同胚的长度空间的最优收缩不等式，通过分析万有覆盖渐近体积增长等方法，推广了黎曼和芬斯勒曲面的经典结果，证明依赖相关理论并基于前人工作。
中文摘要 AI 辅助
我们证明了与亏格为1的环面或实射影平面同胚的长度空间的最优收缩不等式。在这两种情况下，最优常数与（可逆）芬斯勒情形下的常数一致。这推广了黎曼和芬斯勒曲面的经典结果。环面不等式的证明依赖于对万有覆盖渐近体积增长的分析以及二维赋范平面已知面积极小性的加强。实射影平面不等式证明中，类似地扩展了半球的极小性结果。这些结果分别基于Burago - Ivanov和Ivanov的工作。证明中应用了度量圆盘的近期一致化定理以及Lytchak - Wenger的度量空间中面积最小化圆盘理论。
英文摘要
We prove optimal systolic inequalities for length spaces homeomorphic to a torus of genus one or a real projective plane. In both cases, the optimal constant coincides with the constant from the (reversible) Finsler setting. This generalizes the classical results for Riemannian and Finsler surfaces. The proof of the inequality for the torus relies on an analysis of the asymptotic volume growth of the universal cover, together with a strengthening of the previously known area minimality of two-dimensional normed planes. For the inequality of the real projective plane, we similarly extend a minimality result of hemispheres. These results build upon works of Burago-Ivanov and Ivanov, respectively. In their proofs we apply recent uniformization theorems for metric disks and the theory of area-minimizing disks in metric spaces due to Lytchak-Wenger.
