Ancient gradient flows of elliptic functionals and Morse index

Abstract

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or shrinking circles.

Final version; to appear in Amer. J. Math

References in corpus (3)

• Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions
• Uniqueness of asymptotically conical tangent flows
• Classification of compact ancient solutions to the curve shortening flow

Cited by in corpus (14)

• A nonexistence result for wing-like mean curvature flows in ${\mathbb{R}}^4$
• Mean curvature flow with generic initial data
• The atomic structure of ancient grain boundaries
• Construction of High Codimension Ancient Mean Curvature Flows
• Ancient Solution of Mean Curvature Flow in Space Forms
• Ancient finite entropy flows by powers of curvature in ${\mathbb{R}}^2$
• Initial Perturbation of the Mean Curvature Flow for closed limit shrinker
• Classification of ancient flows by sub-affine-critical powers of curvature in ${\mathbb{R}}^2$
• Classification of convex ancient free boundary curve shortening flows in the disc
• Ancient mean curvature flows from minimal hypersurfaces
• The blowdown of ancient noncollapsed mean curvature flows
• Hearing the shape of ancient noncollapsed flows in ${\mathbb{R}}^4$
• Revisiting generic mean curvature flow in ${\mathbb{R}}^3$
• Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$