Mean curvature flow of higher codimension in Riemannian manifolds

Kefeng Liu, Hongwei Xu, Entao Zhao

Published 2012-03-31Version 1

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow the submanifold contracts smoothly to a round point in finite time. As a consequence we obtain a differentiable sphere theorem for submanifolds in a Riemannian manifold.