Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.

We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times \Sigma_2, g_1-g_2)$ of a map $f:\Sigma_1\to \Sigma_2$ from a closed Riemannian manifold $(\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(\Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumptio...

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 priori estimates for a more general class of flows in a local and flexible setting.

In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in $\M^n\times\Real$. Also, for a Riemannian manifold $\M^2$ with negative Gaussian curvature at each point, we show non-existence of complete vertically translating graphs in $\M^2\times\Real$.

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.

Let f:\Sigma_1 --> \Sigma_2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in the product of \Sigma_1 and \Sigma_2 by the mean curvature flow. Under suitable conditions on the curvature of \Sigma_1 and \Sigma_2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map f_t and f_t converges to a constant map as t...

Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant map.

We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time singularities, and constructions of self-similar solutions.

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier condition and a condition on the derivative of prescribed function with respect to the height. As an application we construct a weighted mean curvature flow in large c...

In this paper, we prove that any mean curvature flow translator $\Sigma^2 \subset \mathbb{R}^3$ with finite total curvature and one end must be a plane. We also prove that if the translator $\Sigma$ has multiple ends, they are asymptotic to a plane $\Pi$ containing the direction of translation and can be written as graphs over $\Pi$. Finally, we determine that the ends of $\Sigma$ are strongly asymptotic to $\Pi$ and obtain quantitative estimates for their asymptotic behavior...