致 Alexandrov 几何：CAT(0) 空间

Closed geodesics in Alexandrov spaces of curvature bounded from above

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.

Alexrov空间的Lipschitz同伦收敛II

Lipschitz homotopy convergence of Alexandrov spaces II

本文研究了Alexrov空间的Lipschitz同伦收敛，提出了适用于更广泛情形的同伦稳定性定理，并展示了如何保持奇异性strata的同伦构造。

CD meets CAT

We show that if a noncollapsed $CD(K,n)$ space $X$ with $n\ge 2$ has curvature bounded above by $κ$ in the sense of Alexandrov then $K\le (n-1)κ$ and $X$ is an Alexandrov space of curvature bounded below by $K-κ(n-2)$. We also show that if a $CD(K,n)$ space $Y$ with finite $n$ has curvature bounded above then it is infinitesimally Hilbertian.

Spaces with almost Euclidean Dehn function

We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are ${\rm CAT}(0)$. This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper ${\rm CAT}(0)$ in terms of the growth of the Dehn function at al…

The large scale geometry of nilpotent Lie groups

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an Alexandrov metric space with curvature bounded below. The main technical aspect of this work is the proof of a limited metric differentiability of Lipschitz maps between connected graded nilpotent Lie groups equipped with left invariant Carnot-Caratheodory metrics and complete m…

奇异几何中的调和映射与刚性

Harmonic maps in singular geometry and rigidity

本文综述了调和映射到非正曲率（NPC/CAT(0)）空间的研究，利用非光滑目标的正则性理论，推广了Margulis超刚性并探讨了Teichmüller空间的全纯刚性。