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Generic regularity of Level Set Flows with spherical singularity

Ao Sun, Jinxin Xue

Abstract

The sphere is well-known as the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize the generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, we establish that generically the arrival time function of level set flow with spherical singularity has at most $C^2$ regularity.

Generic regularity of Level Set Flows with spherical singularity

Abstract

The sphere is well-known as the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize the generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, we establish that generically the arrival time function of level set flow with spherical singularity has at most regularity.
Paper Structure (8 sections, 22 theorems, 43 equations)

This paper contains 8 sections, 22 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Spherical singularity
  3. Local dynamics and invariant cones
  4. The stability of slow singularities
  5. Slow singularities are generic
  6. Action of translations and dilations
  7. Denseness of image of the fundamental solution
  8. Transplantation

Key Result

Theorem 1.1

There is an open and dense subset of $\mathcal{G}$, such that the LSF starting from any hypersurface in such a set is $C^2$ but not $C^3$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Theorem 2.1 and (4) in St1
  • Theorem 2.2: Theorem 1.1 in St1
  • Proposition 2.3
  • Theorem 3.1
  • Proposition 3.2: Proposition 5.2 in CM4
  • proof : Proof of Theorem \ref{['thm:invariant cone']}
  • Proposition 3.3: Proposition 3.2 in SX1
  • Proposition 3.4
  • ...and 22 more