Xiaochun Rong, Yusheng Wang
In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.
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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.
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。
Lecture Notes on Comparison Geometry
本比较几何讲义基于多伦多大学相关课程整理，涵盖多种经典比较定理及其应用、多位学者的核心定理与猜想证明，还介绍了Gromov-Hausdorff收敛等进阶内容。
In this paper we discuss the sufficient and necessary conditions for multiple Alexandrov spaces being glued to an Alexandrov space. We propose a Gluing Conjecture, which says that the finite gluing of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to Alexandrov spaces. This generalizes Petrunin's Gluing Theorem. Under the assumptions of the Gluing Conjecture, we classify the $2$-point gluing over $(n-1,ε)$-regular points as local separable gluing and the gluing near un-glued $(n-1,ε)$-regular points a…
Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II
本文肯定回答了半空间凸毛细超曲面 Alexandrov-Fenchel 不等式的猜想，建立毛细凸体理论并证明一般混合体积不等式。
In this note we prove the Borel Conjecture for closed, irreducible and sufficiently collapsed three-dimensional Alexandrov spaces. We also pose several questions related to characterization of fundamental groups of three-dimensional Alexandrov spaces, finite groups acting on them and rigidity results.
Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $Z/\sim$, where $z, z'\in\partial Z$ and $z\sim z'$ if only if $f(z)=f(z')$. We will partially verify this conjecture, and give a classification for compact Alexandrov $n$-spaces with relatively maximal volume. We will also give an elementary proof for a pointed version of Bishop-…
Euler characteristics of collapsing Alexandrov spaces
本研究证明了坍缩Alexandrov空间的欧拉示性数等于极限空间各层紧支集欧拉示性数与对应纤维欧拉示性数的乘积之和，验证了Semyon Alesker的相关猜想。
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…
$SRA$-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…
Decompositions of three-dimensional Alexandrov spaces
本文将三维流形拓扑基本结果推广到Alexandrov 3-空间，证明素分解定理并解决非负曲率闭单连通空间结构猜想，还定义广义Dehn手术，证明闭Alexandrov 3-空间可由S^3中链环手术得到且是四维Alexandrov空间边界。
We prove the reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov-Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh-Khovanskii, Khovanskii-Timorin, Milman-Rotem and R. Schneider on reversed (or complemented) Brunn-Minkowski and Alexandrov-Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly's conject…
We prove that the boundary of an orbit space or more generally a leaf space of a singular Riemannian foliation is an Alexandrov space in its intrinsic metric, and that its lower curvature bound is that of the leaf space. A rigidity theorem for positively curved leaf spaces with maximal boundary volume is also established and plays a key role in the proof of the boundary problem.
Embedded flexible spherical cross-polytopes with non-constant volumes
构造了所有维度球面中的嵌入柔性交叉多胞体，证明其弯曲时体积非常数，为球面多面体风箱猜想提供反例，并提出修正风箱猜想。
Quantitative estimates on the $C^2$-singular sets in Alexandrov spaces
本文针对 Alexandrov 空间中的 $C^2$-奇异点，定义了非负函数 $\mathcal K(x)$ 定量度量其偏离 $C^2$ 的程度，并给出了 Hausdorff 维数与测度的定量估计，推进了 Naber 猜想。
Glued spaces and lower curvature bounds
本文综述光滑与非光滑语境下受下曲率界约束的粘合构造定理并提出猜想，聚焦合成下Ricci曲率界，涉及多种空间类型，为相关研究提供理论梳理与猜想方向。
Stability of optimal transport on metric measure spaces
本文证明了在具有下界里奇曲率限制的度量测度空间上最优传输的稳定性，并推导了Alexandrov空间上的相关结果。
The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces
本文通过将 Lobachevsky 空间单纯形体积视为棱长双曲余弦的函数并研究其解析延拓，证明了奇数维 Lobachevsky 空间中有界柔性多面体满足 Bellows 猜想。
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…
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