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 (J Differ Geom 85(3):357–395, 2010) 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
The mean curvature flow is parabolic and ill-posed backwards in time, ancient solutions are interesting for several reasons
|2 is automatically weakly convex, while a general submanifold satisfying this condition has nonnegative sectional curvature [8], and non-negative curvature operator
Summary
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. Where H (x, t) denotes the mean curvature vector. A solution is referred to as ancient if t0 = −∞. The mean curvature flow is (weakly) parabolic and ill-posed backwards in time, ancient solutions are interesting for several reasons. They arise naturally as tangent flows near singularities [13,17,18,25,26], and are models for singularity profiles of the flow [17]
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