A variational proof of Alexandrov's convex cap theorem

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。

带约束的均曲率流与Alexandrov-Fenchel不等式

A constrained mean curvature flow and Alexandrov-Fenchel inequalities

本文研究了具有毛细边界的星形超曲面的均曲率流，证明了其长期存在性和收敛性，并建立了新的Alexandrov-Fenchel不等式。

半空间中各向异性毛细超曲面的 Alexandrov 定理

Alexandrov's theorem for anisotropic capillary hypersurfaces in the half-space

本文研究半空间中的各向异性常平均曲率嵌入毛细超曲面，借助新的 Heintze-Karcher 不等式与 Minkowski 公式证明其必为截短的 Wulff 形状。

各向异性平均曲率型流与毛细管Alexandrov-Fenchel不等式

Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities

本文研究半空间中星形各向异性ω₀-毛细管超曲面的保体积各向异性平均曲率型流，证明其长时间存在与收敛性，定义广义拟质量积分并建立相关等周与Alexandrov-Fenchel不等式，提供了流方法的新证明视角。

An optimal extension of Perelman's comparison theorem for quadrangles and its applications

In this paper we discuss an extension of Perelman's comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with non-negative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become totally convex relative to a bigger sub-domain. An optimal extension of 2nd variational formula for geodesics by Petrunin will be derived for the case of non-negative curvature. In addition, we also introduced the generalized second fundament forms for subsets in Alexandrov spaces. …

Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Let $X\in\text{Alex}\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\ge -1$. Let the $r$-scale $(k,ε)$-singular set $\mathcal S^k_{ε,\,r}(X)$ be the collection of $x\in X$ so that $B_r(x)$ is not $εr$-close to a ball in any splitting space $\mathbb R^{k+1}\times Z$. We show that there exists $C(n,ε)>0$ and $β(n,ε)>0$, independent of the volume, so that for any disjoint collection $\big\{B_{r_i}(x_i):x_i\in \mathcal S_{ε,\,βr_i}^k(X)\cap B_1, \,r_i\le 1\big\}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates $\mathcal H^k(\m…

半空间中凸毛细超曲面的 Alexandrov-Fenchel 不等式完整族

A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

本文研究半空间中带毛细边界的逆曲率流，证明解保持严格凸并收敛到球冠，进而得到凸毛细超曲面的 Alexandrov-Fenchel 不等式完整族，并发展新的张量最大值原理。

2+1 维平坦辐射时空的 Alexandrov 定理

Alexandrov Theorem for 2+1 flat radiant spacetimes

本文通过改造 Volkov 的变分方法，为 Fillastre 定理提供了构造性证明，并推广了该定理：证明对任意带锥奇点的闭局部欧几里得曲面，存在唯一的局部 Minkowski 3-流形及其中的凸多面体，使其边界与给定曲面等距。

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。

单位球中毛细超曲面的逆曲率流

Inverse curvature flows for capillary hypersurfaces in the unit ball

本文研究了单位欧氏球中严格凸毛细超曲面的逆曲率流，建立了一类该流的存在性与收敛性结果，并作为应用导出了具有自由边界的弱凸超曲面的Alexandrov Fenchel不等式。

Capacity inequalities and rigidity of cornered/conical manifolds

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.

Capillary surfaces of constant mean curvature in a right solid cylinder

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to obtain that the surface is part of a plane or a catenoid.

Capillary surfaces in a cone

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surface and the boundary of the cone. In the particular case that the cone is circular, we prove that the surface is a spherical cap or a planar disc. The proofs are based on an extension of the Alexandrov reflection method by using inversions about spheres.

关于球中凸毛细超曲面的平均曲率型流

On the mean curvature type flow for convex capillary hypersurfaces in the ball

本文研究单位欧氏球中具有毛细边界的凸超曲面的平均曲率型流，证明严格凸初值下解长期存在并光滑收敛到球冠，并由此得到新的Alexandrov-Fenchel不等式。