Ancient solutions of geometric flows with curvature pinching

Abstract

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.

Minor changes, bibliography updated. To appear on The Journal of Geometric Analysis. The final version is available online at https://doi.org/10.1007/s12220-018-0036-0

References in corpus (5)

• Dirichlet sigma models and mean curvature flow
• Uniqueness of closed self-similar solutions to the Gauss curvature flow
• A collapsing ancient solution of mean curvature flow in ${\mathbb{R}}^3$
• The mean curvature flow of submanifolds of high codimension
• Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

Cited by in corpus (4)

• Strong spherical rigidity of ancient solutions of expansive curvature flows
• The atomic structure of ancient grain boundaries
• Ancient Solution of Mean Curvature Flow in Space Forms
• Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery