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.

Generalized Liouville Theorem in Nonnegatively Curved Alexandrov Spaces

In this paper, Yau’s conjecture on harmonic functions in Riemannian manifolds is generalized to Alexandrov spaces. It is proved that the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional and strong Liouville theorem holds in Alexandrov spaces with nonnegative curvature.

Quasi-convex subsets in Alexandrov spaces with lower curvature bound

We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound,which include not only all closed convex subsets without boundary but also all extremal subsets.Moreover,we explore several essential properties of such kind of subsets including a generalized Liberman theorem.It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alexandrov spaces with lower curvature bound.

Radius of locally convex subsets in Alexandrov spaces with curvature ≥1 and radius 〉 π/2

Let X be a complete Alexandrov space with curvature ≥1 and radius 〉 π/2. We prove that any connected, complete, and locally convex subset without boundary in X also has the radius 〉 π/2.

Cyclical Deformations of Quadrangles in S^(2)_(κ)and Defining Properties of Alexandrov Spaces

This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ)space.The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the model 2-plane S^(2)_(κ),including the limit shape of each step.As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on S21.At last we make a contrast of geometric properties of CBA with CBB spaces,including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.

A glance at three-dimensional Alexandrov spaces

We discuss the topology and geometry of closed Alexandrov spaces of dimension three.

Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k+1)×Z.We show that there exists C(n,ε)>0 and𝛽(n,ε)>0,independent of the volume,so that for any disjoint collection{B_(ri)(xi)∶x_(i)∈S_(ε,βri)^(k)(X)∩B_(1),r_(i)≤1,the packing estimateΣr_(i)^(k)≤C holds.Consequently,we obtain the Hausdorff measure estimates H^(k)(S_(ε)^(k)(X))∩B_(1))≤C and H^(n)(B_(r)(S_(ε,βri)^(k))∩B_(1))≤C rn−k.This answers an open question in Kapovitch et al.(Metric-measure boundary and geodesic flow on Alexandrov spaces.arXiv:1705.04767(2017)).We also show that the k-singular set S^(k)(X)=⋃ε>0⋂r>0 S_(ε,r)^(k)𝜖,ris k-rectifiable and construct examples to show that such a structure is sharp.For instance,in the k=1 case we can build for any closed set T⊆S^(1)andε>0 a space Y∈Alex^(3)(0)with S_(ε)^(1)(Y)=Ф(T),whereФ∶S^(1)→Y is a bi-Lipschitz embedding.Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable,1-Cantor set with positive 1-Hausdorff measure.

LIPSCHITZ CONTINUITY OF MINIMIZERS FOR THE GINZBURG-LANDAU FUNCTIONAL BETWEEN ALEXANDROV SPACES

In this paper,we shall prove that any minimizer of Ginzburg-Landau functional from an Alexandrov space with curvature bounded below into a nonpositively curved metric cone must be locally Lipschitz continuous.

REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES

Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this short note,we introduce a notion of p-energy for u which is slightly different from the original definition of Korevaar and Schoen.We show that each minimizing p-harmonic mapping(p≥2)associated to our notion of p-energy is locally Holder continuous whenever its image lies in a compact subset of X.

Alexandrov空间的公理体系

为了研究Alexandrov空间的内部公理体系和序方面的特征,利用点集拓扑学和Locale理论中的已有结论,将各结构限制到Alexandrov空间的框架中,得到Alexandrov空间的等价刻画。研究结果表明Alexandrov空间在范畴意义下同构于Alexandrov邻域系统、Alexandrov闭包算子、Alexandrov内部算子、Alexandrov导算子等,T_0的Alexandrov空间同构于偏序集、对偶等价于完全生成格。Alexandrov空间可以用邻域系统、闭包算子、内部算子、导算子,特殊化序和无点化序进行等价刻画。

空间型中三角形的角关于曲率的某种一致性

给出了单连通空间型中三角形的角关于曲率的某种一致性,即在任意2个具有不同曲率的空间型中,边长对应相等的三角形在其周长趋于0时,它们对应角的比值在某种一致意义下收敛到1.这对研究和理解曲率有界的Alexandrov空间上与角相关的性质是有一定益处的.

双曲空间上的Alexandrov-Fenchel不等式

本文在双曲空间H^n(n≥5)中建立如下的双曲Alexandrov-Fenchel不等式:若超曲面为h-凸的,则有∫∑σ_4dμ≥C_(n-1)~4w_(n-1){(|Σ|/w_(n-1))~1/2)+(|Σ|/w_(n-1))^((1/2)(n-5)/(n-1)}~2等号成立当且仅当Σ为H^n中的测地球面.

Pythagorean Theorem & Curvature with Lower or Upper Bound

In this paper,the authors give a comparison version of Pythagorean theo-rem to judge the lower or upper bound of the curvature of Alexandrov spaces(including Riemannian manifolds).