Gauss maps of the mean curvature flow

Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10064

Mathematical Research Letters, 10, (2), 287-299, 2003
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.
Gauss map, mean curvature flow
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  title={Gauss maps of the mean curvature flow},
  author={Mu-Tao Wang},
  booktitle={Mathematical Research Letters},
Mu-Tao Wang. Gauss maps of the mean curvature flow. 2003. Vol. 10. In Mathematical Research Letters. pp.287-299.
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