Related papers: Mean curvature flows with prescribed singular sets
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components…
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…
It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or…
The symmetry group of the mean curvature flow in general ambient Riemannian manifolds is determined, based on which we define generalized solitons to the mean curvature flow. We also provide examples of homothetic solitons in non-Euclidean…
In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and…
We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in…
We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of…
This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some…
Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$",…
We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a…
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…
We study the flow $M_t$ of a smooth, strictly convex hypersurface by its mean curvature in $\mathrm{R}^{n+1}$. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time $T$ and point $x^*$ (which…
We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of…
In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $\mathbb{R}^N$ in every dimension $N \ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form…
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays…
Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally…
It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the…
The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface,…
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the…
We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity…