A systolic inequality with remainder in the real projective plane

推广 Gromov 的最优收缩不等式

Extending Gromov's optimal systolic inequality

本文将 Gromov 针对复射影空间的最优收缩不等式推广到基本上同调类由二维类的 cup 积构成的流形。

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。

通过稳定性条件研究K3曲面的收缩不等式

Systolic inequalities for K3 surfaces via stability conditions

本文引入Bridgeland稳定性条件的范畴收缩与范畴体积概念，证明任意射影K3曲面上存在仅依赖Picard群秩与判别式的常数C，使收缩平方不超过C倍体积，推广了二维环面经典收缩不等式并讨论其在辛几何中的应用。

A Pu-Bonnesen inequality

We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu's systolic inequality for positively-curved metrics. The remainder term in the inequality, analogous to that in Bonnesen's inequality, is a function of R-r (suitably normalized), where R and r are respectively the circumradius and the inradius of the Weyl-Lewy Euclidean embedding of the orientable double cover. We exploit John ellipsoids of a convex body and Pogorelov's ridigity theorem.

Contact geometry and isosystolic inequalities

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and the use of canonical perturbation theory allow us to solve a parametric version of this problem. Namely, we show that if g_s is a smooth volume-preserving deformation of the canonical metric and at s=0 the deformation is not tangent to all orders to trivial deformations (i.e., to d…

秩1对称空间的校准与能量极小映射

Calibrations and Energy-Minimizing Mappings of Rank-1 Symmetric Spaces

本文证明了从实、复和四元数射影空间到黎曼流形的能量泛函下界，其中实和复情形下界精确，并刻画了能量极小映射族，同时探讨了与脉动几何的联系。