Bi-Lipschitz equivalent Alexandrov surfaces, I

Lipschitz continuity of harmonic maps between Alexandrov spaces

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.

Lipschitz-Volume rigidity on limit spaces with Ricci curvature bounded from below

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

On a new definition of Ricci curvature on Alexandrov spaces

Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.

Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below…

Flat convergence for integral currents in metric spaces

It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in particular all Banach spaces as well as complete CAT(k)-spaces (metric spaces of curvature bounded above by k in the sense of Alexandrov). The main result can be used to prove the existence of minimal elements in a fixed Lipschitz homology class in compact metric spaces admittin…

Harmonic branched coverings and uniformization of CAT($k$) spheres

Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a consequence, if $(S,d)$ is homeomorphically equivalent to the 2-sphere $\mathbb S^2$, then it is conformally equivalent to $\mathbb S^2$.

Alexandrov 空间中的 Allard 内部 ε-正则性定理

Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces

将 Allard 内部 ε-正则性定理推广到具有上下曲率有界的非塌缩 Alexandrov 空间，针对广义平均曲率在 L^p_loc (p>m) 中的 m 维 varifold，通过内在证明和逼近定理实现。

分布曲率界对于低正则性黎曼度量的研究

Distributional sectional curvature bounds for Riemannian metrics of low regularity

本文提出了一种针对低正则性黎曼度量的分布曲率界概念，并证明了其与阿莱克斯安德罗夫空间三角比较的联系。

非负曲率Alexandrov空间上的等周问题与无穷远结构

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

研究非负曲率有限维Alexandrov空间的等周剖面、等周集存在性与无穷远渐近结构的关系，证明线性体积增长等条件的等价性，并给出2维情形下等周集的存在性与拓扑刻画。

On the volume measure of non-smooth spaces with Ricci curvature bounded below

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.

Lagrangian Mean Curvature flow for entire Lipschitz graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\t…

Lipschitz Characterisation of Polytopal Hilbert Geometries

We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.

关于奇异曲面上两种曲率概念

On two notions of curvature on singular surfaces

研究奇异曲面上两种曲率界概念的等价性，通过证明相关不等式蕴含亚历山德罗夫曲率界填补理论空白，并得出结合彼得鲁宁结果，\(\omega\geq\kappa\mu\)意味着\(RCD(\kappa,2)\)的结论。

Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.

Lipschitz 曲面的平方根法向场与 Wasserstein Fisher Rao 度量

Square Root Normal Fields for Lipschitz surfaces and the Wasserstein Fisher Rao metric

本文证明 Lipschitz 曲面上的 SRNF 距离等价于球面 Borel 测度的 WFR 度量，并刻画 SRNF 变换的性质，进而推广 WFR 度量至一般测度空间。