# MathSciDoc: An Archive for Mathematician ∫

#### Differential Geometrymathscidoc:1608.10058

Communications in Analysis and Geometry , 13, (5), 2005
Let $M = \sum_1 \times \sum_2$ be the product of two compact Riemannian manifolds of dimension $n \ge 2$ and two, respectively. Let $\sum$ be the graph of a smooth map $f: \sum_1 \to \sum_2$, then $\sum$ is an n-dimensional submanifold of $M$. Let $\mathcal{G}$ 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 $\gamma :\sum \to \mathcal{G}$ assigns to each point $x \in \sum$ the tangent space of \sum$at x. This article considers the mean curvature flow of$\sum $in$M$. When$\sum_1|$and$\sum_2|$are of the same non-negative curvature, we show a sub-bundle \mathcal{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $\sum$ lies in \mathcal{S}$, then the Gauss map of$\sum_t$at any later time$t$remains in$\mathcal{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.
Riemannian manifolds, Grassmannian bundle
@inproceedings{mu-tao2005subsets,
title={Subsets of Grassmannians preserved by mean curvature},
author={Mu-Tao Wang},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821232558564188372},
booktitle={Communications in Analysis and Geometry },
volume={13},
number={5},
year={2005},
}

Mu-Tao Wang. Subsets of Grassmannians preserved by mean curvature. 2005. Vol. 13. In Communications in Analysis and Geometry . http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821232558564188372.