Optimal systolic inequalities on Finsler Mobius bands

莫比乌斯带与克莱因瓶收缩不等式稳定性

Stability of systolic inequalities for the Möbius strip and Klein bottle

本文研究莫比乌斯带和克莱因瓶上收缩面积不等式的稳定性，给出Bavard结果的替代证明，并用共形因子到极小度量的L²距离估计收缩缺陷。

A Finsler counterexample to the Croke conjecture for the systolic ratio on the 2-sphere

We exhibit a Finsler metric on the 2-sphere whose systolic (Holmes-Thompson) ratio is $\frac{4π}{3}$. This is bigger than the conjectured maximal Riemannian systolic ratio of $2\sqrt{3}$ achieved by the Calabi-Croke metric. The construction of the Finsler metric is heavily inspired by a paper of Cossarini-Sabourau.

Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $π$ and equals $π$ if and only if the metric is Riemannian and Zoll.

The Schwarzian derivative and conformal transformation on Finsler manifolds

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore…

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

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。

Sharp systolic inequalities for Reeb flows on the three-sphere

The systolic ratio of a contact form $α$ on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where $T_{\min}(α)$ is the minimal period of closed Reeb orbits on $(S^3,α)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $ρ_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a…