The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensions were proved. We shall explain the results obtained as well as the techniques involved. The potential applications in symplectic topology and mirror symmetry will also be discussed.

Let $F: \sum^n \times [0, T) \to \mathbb{R}^{n+m} be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $\gamma : (\sum^n, g_t) \to G(n,m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on $G(n,m)$ produces a subsolution of the nonlinear heat equation on $(\sum, g_t)$. We also show the condition that the image of the Gauss map lies in a totally geodesic submanifold of $G(n,m)$ is preserved by the mean curvature flow. Since the space of Lagrangian subspaces is totally geodesic in $G(n, n)$, this gives an alternative proof that any Lagrangian submanifold remains Lagrangian along the mean curvature flow.

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 $df$.