Nonpositively curved 2-complexes with isolated flats

局部椭圆作用、挠群与非正曲率空间

Locally elliptic actions, torsion groups, and nonpositively curved spaces

猜想有限生成群在有限维非正曲率复形上的局部椭圆作用具有全局不动点，并以Helly图为主要工具，针对欧几里得建筑、Helly复形等一大类复形证明了该猜想。

Observable concentration of mm-spaces into nonpositively curved manifolds

The measure concentration property of an mm-space $X$ is roughly described as that any 1-Lipschitz map on $X$ to a metric space $Y$ is almost close to a constant map. The target space $Y$ is called the screen. The case of $Y=\mathbb{R}$ is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}, \cite{milsch}, \cite{sch}, \cite{tal}, \cite{tal2} and its reference). M. Gromov developed the theory of measure concentration in the case where the screen $Y$ is not necessarily $\mathbb{R}$ (cf. \cite{gromovcat}, {gromov2}, \cite{gromov}). In this paper, we consider the case …

Rigidity of nonpositively curved manifolds with convex boundary

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv a$ on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces,…