Subsets of Grassmannians Preserved by Mean Curvature Flows

Abstract

Let $M=\Sigma_1\times \Sigma_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2 $ and two, respectively. Let $\Sigma$ be the graph of a smooth map $f:\Sigma_1\mapsto \Sigma_2$, then $\Sigma$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ 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:\Sigma\mapsto \frak{G} $ assigns to each point $x\in \Sigma$ the tangent space of $\Sigma$ at $x$. This article considers the mean curvature flow of $\Sigma$ in $M$. When $\Sigma_1$ and $\Sigma_2$ are of the same non-negative curvature, we show a sub-bundle $\frak{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $\Sigma$ lies in $\frak{S}$, then the Gauss map of $\Sigma_t$ at any later time $t$ remains in $\frak{S}$. 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.