相关论文: Ancient Solution of Mean Curvature Flow in Space F…

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property…

In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold…

We prove a local version of the noncollapsing estimate for mean curvature flow. By combining our result with earlier work of X.-J. Wang, it follows that certain ancient convex solutions that sweep out the entire space are noncollapsed.

The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean…

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature…

We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we…

We construct an $I$-family of ancient graphical mean curvature flows over a minimal hypersurface in $\mathbb{R}^{n+1}$ of finite total curvature with the Morse index $I$ by establishing exponentially fast convergence in terms of $|x|^2-t$.…

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of…

In this note we construct an infinite family of ancient solutions to the Curve Shortening Flow which span the halfplane.

We construct an embedded ancient solution to the mean curvature flow which is qualitatively given by a ``stack'' of two ancient pancakes joined by a neck. Our solution is closed, non-convex, and contained in a slab.

We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This…

There is an extensive and growing body of work analyzing convex ancient solutions to Mean Curvature Flow (MCF), or equivalently of Rescaled Mean Curvature Flow (RMCF). The goal of this paper is to complement the existing literature, which…

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are…

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and we give precise asymptotic description…

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the…

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive…

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions…

We prove that there exists, in every dimension, a unique (modulo rotations about the origin and time translations) convex ancient mean curvature flow in the ball with free boundary on the sphere.

In this paper, we introduce a new constrained mean curvature type flow for capillary boundary hypersurfaces in space forms. We show the flow exists for all time and converges globally to a spherical cap. Moreover, the flow preserves the…

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean…