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Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

Jiuzhou Huang

TL;DR

This work advances the theory of mean curvature flow by constructing a family of flows that, after type I rescaling, converge locally smoothly to a regular minimizing, strictly stable hypercone ${\mathcal C}$ in $\mathbb{R}^{n+1}$, and after type II rescaling converge to the Hardt–Simon foliation $S_{\kappa,+}$ tangent to ${\mathcal C}$ at infinity. The analysis handles non-symmetric cones by developing a PDE- (not ODE-) framework, leveraging a coercivity estimate for the linearized Jacobi operator, a spectral decomposition, and barrier constructions tied to the Hardt–Simon foliation. A degree-method construction couples spectral data with carefully designed initial perturbations to produce an admissible flow, with detailed $C^0$ and higher-order estimates across outer, intermediate, and inner regions. The results generalize prior symmetric-cone results (Velázquez, Liu) to the non-symmetric setting and illuminate the singularity models and foliation limits for MCF in higher dimensions, underscoring the role of cone minimizing and strict stability assumptions.

Abstract

In this paper, we construct a family of mean curvature flow which converges to an area minimizing, strictly stable hypercone $\mC$ after type I rescaling, and converges to the Hardt-Simon foliation of the cone after a type II rescaling provided the cone satisfies some technique conditions. The difference from Velázquez's previous results is that we drop the symmetry condition on the cone.

Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

TL;DR

This work advances the theory of mean curvature flow by constructing a family of flows that, after type I rescaling, converge locally smoothly to a regular minimizing, strictly stable hypercone in , and after type II rescaling converge to the Hardt–Simon foliation tangent to at infinity. The analysis handles non-symmetric cones by developing a PDE- (not ODE-) framework, leveraging a coercivity estimate for the linearized Jacobi operator, a spectral decomposition, and barrier constructions tied to the Hardt–Simon foliation. A degree-method construction couples spectral data with carefully designed initial perturbations to produce an admissible flow, with detailed and higher-order estimates across outer, intermediate, and inner regions. The results generalize prior symmetric-cone results (Velázquez, Liu) to the non-symmetric setting and illuminate the singularity models and foliation limits for MCF in higher dimensions, underscoring the role of cone minimizing and strict stability assumptions.

Abstract

In this paper, we construct a family of mean curvature flow which converges to an area minimizing, strictly stable hypercone after type I rescaling, and converges to the Hardt-Simon foliation of the cone after a type II rescaling provided the cone satisfies some technique conditions. The difference from Velázquez's previous results is that we drop the symmetry condition on the cone.

Paper Structure

This paper contains 26 sections, 26 theorems, 412 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A coercivity lemma
  4. Functional analysis on ${\mathcal{C}}$
  5. Admissible flows
  6. Constructions of the flow
  7. Initial values
  8. Degree method
  9. $C^0$ estimates
  10. Intermediate region
  11. The compact outer region
  12. Inner region
  13. Choice of $\vartheta,l$
  14. Sub solutions
  15. Upper barrier
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $n\geq 7$ be an integer, ${\mathcal{C}}\subset \mathbb R^{n+1}$ be a regular minimizing, strictly stable hypercone with isolated singularity. Let $\mathcal{L}_{{\mathcal{C}}}$ be the Jacobi operator, $S_+$ be the Hardt-Simon foliation. If $\mathcal{L}_{{\mathcal{C}}}$ has eigenvalue $\lambda_l$ as $s\to\infty$. Moreover, the type II rescaled hypersurfaces $\{\Gamma_\tau\}_{\tau_0\leq \tau<\in

Theorems & Definitions (59)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Lemma 4.4
  • proof
  • ...and 49 more