Gauss Maps of the Mean Curvature Flow

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

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Published 1 January 2003