Mean curvature flow of higher codimension in Riemannian manifolds

Abstract

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.

28 pages

References in corpus (2)

• The extension and convergence of mean curvature flow in higher codimension
• Mean Curvature Flow of Higher Codimension in Hyperbolic Spaces

Cited by in corpus (4)

• An Optimal Convergence Theorem for Mean Curvature Flow of Arbitrary Codimension in Hyperbolic Spaces
• New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory
• A New Version of Huisken's Convergence Theorem for Mean Curvature Flow in Spheres
• Ancient Solution of Mean Curvature Flow in Space Forms