A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite ...

In this note we provide examples of compact embedded lagrangians in Cn for any n ≥ 2 that under mean curvature flow develop singularities in finite time. When n is odd the lagrangians can be taken to be orientable. By gluing these lagrangians onto a special lagrangian embedding L we provide examples of compact embedded lagrangians in a Calabi-Yau manifold that under mean curvature flow develop ...

It was proved that a blow-up solution to the mean curvature flow with positive mean curvature is an ancient convex solution, that is a convex solution which exists for time t from −∞. In this paper we study the geometry of ancient convex solutions. Our main results are contained in Theorems 1-3 below. Theorem 1 asserts that after normalization, the solution converges to a sphere or cylinder as ...

We study the evolution of grain boundary networks by the mean-curvature flow under the restriction that the networks are Voronoi diagrams for a set of points. For such evolution we prove a rigorous universal upper bound on the coarsening rate. The rate agrees with the rate predicted for the evolution by mean-curvature flow of the general grain boundary networks, namely that the typical grain ar...

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an L bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regularity theorem of Ecker and Huisken for Lipschitz hypersurfaces. In particular, any ...

In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a “neck” which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.

We study the motion of an n-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power p of the mean curvature. We prove that for any p ∈ (0, 1], the flow exists for all time when the Ricci tensor of the ambient space is bounded from below on the set of timelike unit vectors. Moreover, if we ass...

In this work, it is demonstrated that the existence and topology of the recently observed interfacial singularities driven by Marangoni effects can be deduced using mean-curvature flow theory extended to account for variations of interfacial tension. This suggests that some of the physical mechanisms underlying the formation of these interfacial singularities may originate from/be modeled by th...

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is C3close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.