允许点态对称的仿射超曲面

允许超曲面法向移位的动力系统理论中的第一个整体化问题

First problem of globalization in the theory of dynamical systems admitting the normal shift of hypersurfaces

考虑了黎曼流形中允许超曲面法向位移的牛顿动力系统的力场公式。研究了与该公式相关的几何结构的全局化问题。

容许超曲面法向移位的动力系统理论中的第二个整体化问题

Second problem of globalization in the theory of dynamical systems admitting the normal shift of hypersurfaces

考虑了允许法向位移的动力系统理论中几何结构的整体积分问题。在这种整合是可能的情况下，转移地图的全球化问题进行了研究。

在允许非平凡共形向量场的黎曼流形中的等周问题

The isoperimetric problem in the Riemannian manifold admitting a non-trivial conformal vector field

本文在具有非平凡共形向量场的黎曼流形中引入保体积、降面积的平均曲率型流，证明其长时间存在与收敛，并建立星形区域等周不等式。

超曲面法向注入半径下界及带有边界的流形上有界几何的构造

Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary

本文首次对超曲面法向注入半径进行点wise下界估计，并在任意带有边界的流形上构造具有有界几何的度量，同时证明边界可变为凸。

五维常曲率空间中极小旋转超曲面的一个刚性结果

A Rigidity Result for Minimal Rotation Hypersurfaces of 5D Spaces of Constant Curvature

本文证明五维常曲率空间中极小超曲面上的一个外在逐点不变量非正，且其恒为零当且仅当超曲面为旋转超曲面，并猜想该刚性结果可推广到其他维数。

Real hypersurfaces with $^{*}$-Ricci solitons of non-flat complex space forms

Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex space form admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.

Evolution of non-compact hypersurfaces by inverse mean curvature

We study the evolution of complete non-compact convex hypersurfaces in $\mathbb{R}^{n+1}$ by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a'priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof o…

零超曲面数据与环境向量场：零阶/一阶 Killing 视界

Null hypersurface data and ambient vector fields: Killing horizons of order zero/one

本文研究带切向零向量场 η 的零超曲面，建立其形变张量与几何张量的恒等式，推广表面引力并提出三类抽象视界，涵盖一般拓扑与不动点情形。

Every centroaffine Tchebychev hyperovaloid is ellipsoid

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding pr…

欧几里得空间中稳定极小超曲面的格林核梯度估计及$\mathbb{R}^4$中的稳定伯恩斯坦定理

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

本文通过新积分不等式证明了稳定极小超曲面的性质，并给出$\mathbb{R}^4$中完整双侧稳定极小超曲面是平面的简单证明，核心方法基于非负曲率空间中稳定极小超曲面具有谱里奇曲率下界。

Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem

Using minimal hypersurfaces, we obtain topological obstructions to admitting complete metrics with positive scalar curvature on a given class of non-compact n-manifolds with n less than 8. We show that the Liouville theorem for a locally conformally flat n-manifold of non-negative scalar curvature follows from the impossibility of there being a positive scalar curvature metric on its connect sum with the n-torus. With the recent work of Chodosh-Li, the Liouville theorem is now proved in all remaining cases. Finally, using MOTS instead of minimal hypersurfaces, we show an Initial Data Set versi…