非紧自收缩子的一个非打结性结果

Geometry of Two-dimensional Self-shrinkers

We prove a local graphical theorem for two-dimensional self-shrinkers away from the origin. As applications, we study the asymptotic behavior of noncompact self-shrinkers with finite genus. Also, we show uniform boundedness on the second fundamental form of two-dimensional noncompact self-shrinkers with bounded mean curvature and uniform locally finite genus.

Lower volume growth estimates for Self-shrinkers of mean curvature flow

We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.

Kähler–Ricci 收缩子的标量曲率界与切锥

Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers

本文证明了 Kähler–Ricci 收缩子的标量曲率仅依赖维数的一致界，并建立了非紧收缩子无穷远处唯一切锥与其极化 Fano 纤维化基的同胚关系。

Asymptotic structure of self-shrinkers

We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.

Self-shrinkers of the mean curvature flow in arbitrary codimension

For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution in arbitrary codimension and of dimension greater than one is spherical, i.e. contained in a sphere, if and only if the mean curvature vector \be H\ee is non-vanishing and the principal normal \beν\ee is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.

闵可夫斯基空间中的逆黑塞曲率流 II：无穷远狄利克雷问题

Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity

本文研究闵可夫斯基空间中非紧致类空严格凸超曲面的逆σ_k曲率流，证明其存在整体解且归一化后收敛到对应唯一自收缩子，丰富了非紧致情形下自收缩子的相关理论。

非紧自收缩子上的对数 Sobolev 不等式

The Logarithmic Sobolev inequality on non-compact self-shrinkers

本文利用 ABP 方法将 Brendle 关于闭自收缩子的最优对数 Sobolev 不等式推广到欧氏空间中完备、非紧、适当嵌入的自收缩子，得到 Ecker 结果的尖锐版本。

Kähler-Ricci收缩子的Matsushima定理、唯一性与K-多重稳定性

A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers

本文针对非紧Kähler-Ricci收缩子证明了Matsushima定理、双全纯唯一性及极化Fano纤维化的K-多重稳定性。

$κ$-solutions 以圆柱体为渐近收缩解

$κ$-solutions with the round cylinder as an asymptotic shrinker

研究显示，n≥4维Ricci流$κ$-解，其渐近收缩Ricci溶菌为圆柱体，必须均匀PIC。结合前期分类结果，此类非紧解要么为圆柱体收缩解，要么为Bryant稳态解；紧解则为Perelman古典型解。

具有渐近锥形奇点的平均曲率流的一般动力学

Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

该论文作为平均曲率流一般动力学系列的第二篇，针对首次奇点为渐近锥形收缩子的平均曲率流初始扰动展开研究，引入费曼-卡茨公式和非紧不变锥方法，证明一般初始扰动可使重标流避免锥形奇点。

A pointwise approach to rigidity of almost graphical self-shrinking solutions of mean curvature flows

We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.

具有非负Ricci曲率的四维收缩孤立子

Four-dimensional shrinkers with nonnegative Ricci curvature

本文通过Ricci曲率特征值演化渐近分析、带边界Gauss-Bonnet-Chern公式和分部积分，在截面曲率或数量曲率等几何条件下，将具有非负Ricci曲率的四维收缩孤立子分类为 R×S^3 或 R^2×S^2。

No breather theorems for the mean curvature flow

In this article we study the breathers of the mean curvature flow in the Euclidean space. A breather is a solution to the mean curvature flow which repeats itself up to isometry and scaling once in a while. We prove several no breather theorems in the noncompact category, that is, under certain conditions, a breather of the mean curvature flow must be a solitonic solution (self-shrinker, self-expander, or translator).