相关论文: Blow Up of Compact Mean Curvature Flow Solutions w…
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…
In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly…
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower…
We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the…
J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type $\mathrm{II}$ singularity at the origin. He also showed that under a…
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…
In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean…
In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar…
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 "$M(T)$",…
In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various…
We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs…
We study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators…
In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds are characterised by the blow up of the mean curvature.
We consider ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$.…
We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore…
In this paper we study the blow up sequence of mean curvature flow of surfaces in $\mathbb R^3$ with additional forces. We prove that the blow up limit of a mean curvature flow of smoothly embedded surfaces with additional forces with…
In this paper, the authors consider an initial boundary value problem for the heat flow of equation of surfaces with constant mean curvatures, which was investigated in [On the heat flow of equation of surfaces of constant mean curvatures,…
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…
In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$…
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.