Gluing of multiple Alexandrov spaces

On gluing Alexandrov spaces with lower Ricci curvature bounds

In this paper we prove that in the class of metric measure spaces with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD(K,N)$ with $K\in \mathbb{R}$ and $N\in [1,\infty)$ is preserved under doubling and gluing constructions.

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.

A counterexample to gluing theorems for MCP metric measure spaces

Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq κ$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the $\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane, which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.

致 Alexandrov 几何：CAT(0) 空间

An invitation to Alexandrov geometry: CAT(0) spaces

本文通过最少的准备，展示 Alexandrov 几何的美丽与强大，涵盖 CAT(0) 空间的多个应用和定理。

粘合空间与下曲率界

Glued spaces and lower curvature bounds

本文综述光滑与非光滑语境下受下曲率界约束的粘合构造定理并提出猜想，聚焦合成下Ricci曲率界，涉及多种空间类型，为相关研究提供理论梳理与猜想方向。

将带电黑洞粘合进德西特空间

Gluing charged black holes into de Sitter space

本文将Hintz的黑洞粘合方法推广到爱因斯坦-麦克斯韦系统，把多个带电黑洞粘入德西特空间共形边界，给出基于Friedrich共形约束方程的充要条件，并改进构造方法以省去早期规范固定。