Abstract

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.

Highlights

  • In this paper, we consider ancient solutions to the mean curvature flow with pinched second fundamental form

  • A family of smooth immersions F : Mn × (t0,t1) → Rn+k is a solution to the mean curvature flow if

  • We will always assume that n ≥ 2, k ≥ 1, and that Mn is a complete smooth manifold

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Summary

INTRODUCTION

We consider ancient solutions to the mean curvature flow with pinched second fundamental form. This is a natural high codimension generalisation of a theorem of Hamilton [11], which asserts the compactness of complete hypersurfaces satisfying (2). We would like to thank Mat Langford for many helpful discussions which have benefited this work

PRELIMINARIES
Evolution Equations
CURVATURE PINCHING
ANCIENT SOLUTIONS IN EUCLIDEAN SPACE
Convergence to round points
PINCHED ANCIENT SOLUTIONS IN THE SPHERE
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