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.
Huisken, G., Sinestrari, C. (2015). Convex ancient solutions of the mean curvature flow. JOURNAL OF DIFFERENTIAL GEOMETRY, 101(2), 267-287 [10.4310/jdg/1442364652].
Convex ancient solutions of the mean curvature flow
SINESTRARI, CARLO
2015-01-01
Abstract
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.File | Dimensione | Formato | |
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