Convex ancient solutions of the mean curvature flow

By G. Huisken, Carlo Sinestrari. 2014

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We study solutions of the mean curvature flow which are defined for all negative times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter, or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed in a sphere.

Subfields: Applied Mathematics, Geometry and Topology

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