Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

Ricci Curvature on Alexandrov spaces and Rigidity Theorems

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.

Alexandrov空间边界体积的精确界与刚性

Sharp bounds and rigidity for volumes of boundaries of Alexandrov spaces

该研究针对具有给定曲率下界、维度和半径的Alexandrov空间，推导了其边界体积的精确界，完成刚性情形分类与几乎刚性分析，成果对带边光滑流形也具创新性。

Lipschitz-Volume rigidity in Alexandrov geometry

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.

Relatively maximum volume rigidity in Alexandrov geometry

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $Z/\sim$, where $z, z'\in\partial Z$ and $z\sim z'$ if only if $f(z)=f(z')$. We will partially verify this conjecture, and give a classification for compact Alexandrov $n$-spaces with relatively maximal volume. We will also give an elementary proof for a pointed version of Bishop-…

二维Alexandrov空间边界测地曲率的比较与刚性定理

Comparison and Rigidity Theorems for geodesic curvatures in two dimensional Alexandrov spaces

本文研究了二维Alexandrov空间边界的测地曲率，证明了比较和刚性定理，扩展了已知的曲率上界和下界结果。

Rigidity theorems for glued spaces being suspensions, cones and joins in Alexandrov geometry with curvature bounded below

In the paper, we give rigidity theorems when the glued space of two Alexandrov spapces with curvature bounded below is a suspension, cone or join. And we list some basic properties of joins in Appendix.