Related papers: How close is too close for singular mean curvature…

In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces $M^1_t$ and $M^2_t$ encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time $T$, and if $M^1_T=M^2_T$…

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. We…

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. In…

In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the…

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 show that the intrinsic diameter of mean curvature flow in $\mathbb{R}^3$ is uniformly bounded as one approaches the first singular time $T$. This confirms the bounded diameter conjecture of Haslhofer. In addition, we establish several…

In this paper, we produce explicit examples of mean curvature flow of (2m-1)-dimensional submanifolds which converge to (2m-2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in $\mathbb{C}^{m}$…

For every closed set $K \subset \mathbb{R}^n$ and every $m \geq 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in $\mathbb{R}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily…

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity…

We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In…

Let $(M^n,g_0)$ and $(\bar{M}^{n+1},\bar{g})$ be complete Riemannian manifolds with $|\bar{\nabla}^k\bar{Rm}|\le \bar{C}$ for $k \le 2$, and suppose there is an isometric immersion $F_0: M^n \rightarrow \bar{M}^{n+1}$ with bounded second…

In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H…

We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular…

We will consider a {\it $\tau$-flow}, given by the equation $\frac{d}{dt}g_{ij} = -2R_{ij} + \frac{1}{\tau}g_{ij}$ on a closed manifold $M$, for all times $t\in [0,\infty)$. We will prove that if the curvature operator and the diameter of…

In this paper, we prove that if $M_t\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, is the $n$-dimensional closed embedded $\mathcal{F}-$stable solution to mean curvature flow with mean curvature of $M_t$ is uniformly bounded on $[0,T)$ for…

We will consider a {\it $\tau$-flow}, given by the equation $\frac{d}{dt}g_{ij} = -2R_{ij} + \frac{1}{\tau}g_{ij}$ on a closed manifold $M$, for all times $t\in [0,\infty)$. We will prove that if the curvature operator and the diameter of…

In this paper, we study the $k$-cylindrical singular set of mean curvature flow in $\mathbb R^{n+1}$ for each $1\leq k\leq n-1$. We prove that they are locally contained in a $k$-dimensional $C^{2,\alpha}$-submanifold after removing some…

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if…

Let $M\subset {\mathbf R}^{m+1}$ be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial $k^{\rm th}$ homology. We show that the entropy of $M$ is greater than or equal to the entropy of a round…