Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mean curvature flow with multiplicity $2$ convergence in closed manifolds

Jingwen Chen, Ao Sun

TL;DR

The paper addresses higher multiplicity convergence in the mean curvature flow on closed manifolds by constructing explicit immortal flows that converge to a minimal hypersurface with multiplicity $2$. It leverages rotational symmetry and barrier methods, building an interpolation family of initial data bridging Angenent-type barriers and spherical catenoids to elicit long-time behavior. The authors prove the existence of an immortal flow on $S^n \times [-1,1]$ that limits to $S^n \times \{0\}$ with multiplicity $2$, with a convergent, stable limit and controlled gradient decay. These results illuminate non-generic aspects of higher multiplicity in min-max theory and provide explicit examples of high multiplicity limits within closed ambient manifolds, contributing to the understanding of long-time dynamics and minimal surface formation under mean curvature flow.

Abstract

We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.

Mean curvature flow with multiplicity $2$ convergence in closed manifolds

TL;DR

The paper addresses higher multiplicity convergence in the mean curvature flow on closed manifolds by constructing explicit immortal flows that converge to a minimal hypersurface with multiplicity . It leverages rotational symmetry and barrier methods, building an interpolation family of initial data bridging Angenent-type barriers and spherical catenoids to elicit long-time behavior. The authors prove the existence of an immortal flow on that limits to with multiplicity , with a convergent, stable limit and controlled gradient decay. These results illuminate non-generic aspects of higher multiplicity in min-max theory and provide explicit examples of high multiplicity limits within closed ambient manifolds, contributing to the understanding of long-time dynamics and minimal surface formation under mean curvature flow.

Abstract

We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity as time approaches infinity.
Paper Structure (11 sections, 19 theorems, 63 equations, 3 figures)

This paper contains 11 sections, 19 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Metric on Lg and rotationally symmetric mean curvature flows
  4. Minimal hypersurfaces as barriers
  5. Lg-Angenent curves
  6. Main result
  7. Rotationally symmetric mean curvature flow
  8. Interpolation family of hypersurfaces
  9. Multiplicity Lg convergence
  10. Curves in the plane with conformal metrics
  11. Lg-Angenent curves

Key Result

Theorem 1.1

For all $n\geq 2$, there exists a smooth embedded connected immortal mean curvature flow $(M(t))_{t>0}$ in $S^n\times[-1,1]$ such that $M(t)$ converges to $S^n\times\{0\}$ with multiplicity $2$ as $t\to\infty$.

Figures (3)

  • Figure 1: Example of a section curve that we study.
  • Figure 2: The family of interpolation curves.
  • Figure 3: The blue curves are the flows that we want to study. The red curve on the left represents one $\lambda$-Angenent curve; the purple curve on the right represents one spherical catenoid.

Theorems & Definitions (39)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Proposition 2.1: Comparison principle
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 29 more