Shin-ichi Ohta
We prove that Alexandrov spaces $X$ of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of $X$ into a 2-uniformly convex Banach space is extended as a Lipschitz continuous map on the entire space $X$.
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In this paper, we study a complete noncompact nonnegatively curved Alexandrov space $A$ with a soul $S$ of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in this case Sharafutdinov retraction, $π:\ A\to S$, is a submetry.
We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $\mathbb{R}P^2$. We then classify closed three-dimensional $\mathsf{CD}^*(0,3)$-Alexandrov spaces.
The Optimal Scaling Parameter in Metric Cotype for Alexandrov Spaces of Nonnegative Curvature
本文确定了非负曲率 Alexandrov 空间中度量余型 2 不等式缩放参数的最优阶为线性，并回答了相关开放问题。
Alexandrov spaces with non negative curvature and displacement convexity of the entropy tensor
将熵张量推广到曲率有下界的Alexandrov空间，通过构造满足上循环条件和第二变分公式的平行平凡化，证明非负曲率等价于熵张量的矩阵位移凸性。
It is shown that convex hypersurfaces in Hilbert spaces have nonnegative Alexandrov curvature. This extends an earlier result of Buyalo for convex hypersurfaces in Riemannian manifolds of finite dimension.
In a previous paper Hua-Jost-Liu, we have applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincaré inequality on these graphs to obtain an dimension estimate of polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on graphs, we directly argue on Alexandrov spaces to obtain the optimal dimension estimate of polynomial growth harmonic functions on graphs which is actually linear in th…
The Injectivity Radius of Souls of Alexandrov Spaces
本文将Šarafutdinov关于非紧非负曲率黎曼流形注入半径的下界推广到Alexandrov空间，证明当空间注入半径不等于其灵魂的注入半径时，其至少为$πK^{-1/2}$，并系统引入灵魂概念及比较两种注入半径定义。
We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.
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。
Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery) Ricci curvature.
Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $\mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,\mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,\mathcal{H}^n)$ has finite topological type.
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karcher's classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.
We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.
We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of the real projective plane; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincaré Conjectur…
We give a new characterization of spaces with nonnegative curvature in the sense of Alexandrov.
In the present paper, we apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature in the sense of Higuchi. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincaré inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. In addition, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.
$SRA$-free spaces is a wide class of metric spaces including finite dimensional Alexandrov spaces of non-negative curvature, complete Berwald spaces of nonnegative flag curvature, Cayley Graphs of virtually abelian groups and doubling metric spaces of non-positive Busemann curvature with extendable geodesics. This class also includes arbitrary big balls in complete, locally compact $CAT(k)$-spaces $(k \in \mathbb R)$ with locally extendable geodesics, finite-dimensional Alexandrov spaces of curvature $\ge k$ with $k \in R$ and complete Finsler manifolds satisfying the doubling condition. We sh…
On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here is optimal, and hence answers an open question of Funano. This approach works still on a compact finite-dimensional Alexandrov space of nonnegative curvature and proves affirmatively a conjecture of Funano and Shioya asserting a dimension free upper bound for eigenvalue ratios i…
This article settles the convergence question for multivariate barycentric subdivision schemes with nonnegative masks on complete metric spaces of nonpositive Alexandrov curvature, also known as Hadamard spaces. We establish a link between these types of refinement algorithms and the theory of Markov chains by characterizing barycentric subdivision schemes as nonlinear Markov semigroups. Exploiting this connection, we subsequently prove that any such scheme converges on arbitrary Hadamard spaces if and only if it converges for real valued input data. Moreover, we generalize a characterization …
We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is $q$-barycentric for some $q\in [1,\infty)$ has metric cotype $q$ with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banac…
Equipped with the L^2-distortion distance, the space "X" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on "X" are presented.
Nonnegative cross-curvature in infinite dimensions: synthetic definition and spaces of measures
本文提出非负交叉曲率在无穷维和非光滑情形下的变分定义，证明该类空间在 Gromov-Hausdorff 收敛下封闭，并推广经典性质，特别证明 Wasserstein 空间继承 NNCC 性质，同时给出多种距离作为 NNCC 成本的实例。
Self-contracted curves in spaces with weak lower curvature bound
本文引入弱下曲率界条件，证明相应度量空间中有界自收缩曲线可求长，并建立大雪花不可嵌入性，将小粗糙角策略推广至下曲率界空间。
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