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Mean curvature ows of hypersurfaces have been extensively studied and there are various dierent approaches and many beautiful results. However, relatively little is known about mean curvature ows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature ows in general dimensions and co-dimensions. The related techniques in the so called \blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature ows of two-dimensional surfaces in four-dimensional ambient spaces.

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.

We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

Let $M = \sum_1 \times \sum_2 $ be the product of two compact Riemannian manifolds of dimension $n \ge 2$ and two, respectively. Let $\sum $ be the graph of a smooth map $f: \sum_1 \to \sum_2$, then $\sum$ is an n-dimensional submanifold of $M$. Let $\mathcal{G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all n-dimensional subspaces of the tangent space of $M$. The Gauss map $\gamma :\sum \to \mathcal{G}$ assigns to each point $x \in \sum $ the tangent space of \sum$ at x. This article considers the mean curvature flow of $\sum $ in $M$. When $\sum_1|$ and $\sum_2|$ are of the same non-negative curvature, we show a sub-bundle \mathcal{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $\sum $ lies in \mathcal{S}$, then the Gauss map of $\sum_t$ at any later time $t$ remains in $\mathcal{S}$. We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity . As an application, we show that if $f$ is any map from $S^n$ to $S^2$ and if at each point, the restriction of $df$ to any two dimensional subspace is area decreasing, then f is homotopic to a constant map.

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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 isotopic to a constant map.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the 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 submanifold of the Euclidean space with a continuous induced metric can be smoothed out by the mean curvature flow. The smallness assumption is necessary in the higher codimension case in view of an example of Lawson and Osserman. The stationary phase of the mean curvature flow corresponds to minimal submanifolds. Our result thus generalizes Morrey’s classical theorem on the smoothness of $C_1$ minimal submanifolds.

Let $\Omega $ be a bounded $C^2$ domain in $R^n$ and $\phi : \partial \Omega \to \mathbb{R}^m$ be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map $f : \Omega \to \mathbb{R}^m$ with $ f |_{\partial \Omega}$ and with the graph of $f$ a minimal submanifold in $R^{n+m}$. For $m = 1$, the Dirichlet problem was solved more than thirty years ago by Jenkins-Serrin [13] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension $m$. We prove if $\psi : \bar{\Omega} \to \mathbb{R}^m$ satisfies $8n\delta sup_{\Omega}||D^2 \psi| + \sqrt{2} sup_{\partial \Omega} |D\psi| <1$, then the Dirichlet problem for $ \psi |_{\partial \Omega}$ is solvable in smooth maps. Here $\delta$ is the diameter of $\Omega$ . Such a condition is necessary in view of an example of Lawson-Osserman [15]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with $\psi$ as initial data.

An n dimensional minimal submanifold $\sum$  of $R^{n+m}$ is called nonparametric if $\sum$  can be represented as the graph of a vector-valued function $f : D \subset \mathbb{ R}^n \to \mathbb{R}^n$. This note provides a sufficient condition for the stability of such $\sum$ in terms of the norm of the differential $df$.