关于具有多个端的自我收缩子的解结结果

Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete properly embedded self-shrinker asymptotic to it.

Ricci 收缩子的端部的几何结构

Geometric Structure of Ends of Ricci Shrinkers

本文在无全局曲率假设下研究 Ricci 收缩子端部的几何结构，证明标量曲率满足 Type I 界的点处其 F-极限分裂出直线，并扩展了 F-收敛理论。

On the finiteness of the Morse index of self-shrinkers

In this paper, we present a sufficient condition for finite Morse index of complete properly self-shrinkers. We prove that a complete properly embedded self-shrinker in $\mathbb{R}^{n+1}$ with finite asymptotically conical ends or asymptotically cylindrical ends must have finite Morse index. Moreover, as an application of this result, we show that a complete properly embedded self-shrinker in $\mathbb{R}^3$ with finite genus has finite Morse index.

通过堆叠 $\mathbb{R}^{2}$ 在 $\mathbb{R}^{3}$ 中构造具有任意多个端的自收缩子

Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$

本文利用 PDE 粘合与 LD 方法，通过多层平面和悬链线型桥构造了 $\mathbb{R}^3$ 中具有任意端数、指定亏格及棱柱或反棱柱对称性的自收缩子。

Classification and Analysis of Mean Curvature Flow Self-Shrinkers

We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form $\mathbb S^k\times\R^{n-k}\subset \R^{n+1}$. We use a connection between the stability operator and the quantum harmonic oscillator Hamiltonian to find all eigenvalues and eigenfunctions of the stability operator on these self-shrinkers. We also show self-shrinkers of this form have lower index than all other complete self-shrinking hypersurfaces. In particular, they have finite index. This implies that the ends…

任意亏格的平均曲率流非紧自收缩子

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

本文利用 min-max 方法严格证明了 Ilmanen 关于任意亏格非紧自收缩子存在的猜想，并在大亏格情形下证实其恰有一个渐近锥形端，同时给出奇数亏格双端情形的数值证据。

渐近锥增厚的自收缩子

Self-shrinkers whose asymptotic cones fatten

本文用变分方法构造了熵小于2、具有棱柱对称群的高亏格自收缩子，并利用它们作为初始条件证明了平均曲率流在首个奇异时间后存在非唯一演化，从而证实了Angenent-Chopp-Ilmanen的猜想。

Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone.

Self-shrinkers with a rotational symmetry

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $Σ^n\subseteq\mathbb{R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\mathbb{R}^{n+1}$, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE. We also prove the following classification result: a given complete, embedded, self-shrinking hypersur…

平均曲率流中的增厚现象

Fattening in mean curvature flow

本文证明了在平均曲率流中，对于每个亏格$g\ge 3$均存在紧致曲面$M_g$，其在首次奇异时刻恰有一个奇异点且切向流为特定收缩子；当$g$充分大时$M_g$会发生增厚现象，且随$g$趋于无穷大收缩子收敛于重数为2的平面。