Gauss Maps of the Mean Curvature Flow

Abstract

Let $F:\Sigma^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $\gamma:(\Sigma^n, g_t)\to G(n, m)$ 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 $(\Sigma, g_t)$. 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.