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Rotational symmetry of ancient solutions to fully nonlinear curvature flows

A. Cogo, S. Lynch, O. Vičánek Martínez

Abstract

We address the classification of ancient solutions to fully nonlinear curvature flows for hypersurfaces. Under natural conditions on the speed of motion we classify ancient solutions which are convex, noncollapsing, uniformly two-convex and noncompact. There are exactly two possibilities -- every such solution is either a self-similarly shrinking cylinder, or else is a rotationally symmetric translating soliton. For a large class of flows this yields a complete classification of the blow-up limits that can arise at a singularity of a solution which is compact, embedded and two-convex.

Rotational symmetry of ancient solutions to fully nonlinear curvature flows

Abstract

We address the classification of ancient solutions to fully nonlinear curvature flows for hypersurfaces. Under natural conditions on the speed of motion we classify ancient solutions which are convex, noncollapsing, uniformly two-convex and noncompact. There are exactly two possibilities -- every such solution is either a self-similarly shrinking cylinder, or else is a rotationally symmetric translating soliton. For a large class of flows this yields a complete classification of the blow-up limits that can arise at a singularity of a solution which is compact, embedded and two-convex.
Paper Structure (22 sections, 69 theorems, 420 equations)

This paper contains 22 sections, 69 theorems, 420 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Overview of the proof
  4. Preliminaries
  5. Uniform ellipticity
  6. Compactness of noncollapsing convex ancient solutions
  7. Neck-cap decomposition
  8. Differential Harnack inequality
  9. Expansions for hypersurfaces close to a cylinder
  10. Asymptotic cones
  11. The parabolic blow-down limit
  12. A barrier construction
  13. A priori estimates
  14. Local existence
  15. Global behaviour
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $M_t$, $t \in (-\infty, T]$, be a convex ancient $G$-flow which is noncollapsing and uniformly two-convex. Suppose also that $M_t$ is noncompact. Then $M_t$ is either a self-similarly shrinking round $\mathbb{R}\times S^{n-1}$, or else is a rotationally symmetric translating soliton.

Theorems & Definitions (133)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 123 more