Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities

Rigidity of generic singularities of mean curvature flow

Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, [CM1] showed that the only generic are round cylinders $\SS^k\times \RR^{n-k}$. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder. To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas deve…

叶状 Plateau 问题、曲面 Radon 变换与三维边界面积刚性

Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three

本文研究三维黎曼球上由椭圆曲率泛函定义的 $\Phi$-曲面，通过求解叶状 Plateau 问题构造单位切丛的叶状结构，分析曲面 Radon 变换的核与单射性，并应用于极小曲面叶状结构下的边界面积刚性问题的解决。

具有给定奇点的能量最小化环面值映射、Plateau 问题与 BV 提升

Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting

本文建立余维数为一时环面值映射的 $W^{1,1}$ 能量、BV 提升能量与 normal/integral 流 Plateau 最小质量之间的等价关系，并给出积分 Plateau 问题解的界。

Rigidity of singularities of 2D gravity water waves

We consider the Cauchy problem for the 2D gravity water wave equation. Recently Wu \cite{Wu15, Wu18} proved the local well-posedness of the equation in a regime which allows interfaces with angled crests as initial data. In this work we study properties of these singular solutions and prove that the singularities of these solutions are "rigid". More precisely we prove that an initial interface with angled crests remains angled crested, the Euler equation holds point-wise even on the boundary, the particle at the tip stays at the tip, the acceleration at the tip is the one due to gravity and th…

Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

五维定常Navier-Stokes流的临界Morrey刚性与可去奇点

Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows

该研究证明了五维定常Navier-Stokes方程的临界Morrey刚性定理，推导了五维可去奇点准则，还通过有限能量论证证明了四维对应的三次Morrey刚性定理。