Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of the volume preserving fractional mean curvature flow for convex sets

Vesa Julin, Domenico Angelo La Manna

TL;DR

The paper proves that the volume-preserving fractional mean curvature flow for convex initial sets remains smooth for all time and converges exponentially to a ball. It achieves this by first obtaining uniform $C^{1+s}$-regularity from convexity-based curvature bounds, then deriving a height-function formulation and upgrading regularity to $C^{2+s+\alpha}$ via sphere-adapted parabolic Schauder estimates. The key technical advance is a robust $C^{2+\alpha}$ regularity step for the flow that does not rely on convexity for the final stage and leverages a quasilinear formulation of the differentiated flow. Consequently, the flow does not develop singularities and, by existing results, converges exponentially fast to a translated ball, extending Huisken-type convergence to a nonlocal, fractional setting with a solid regularity framework.

Abstract

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga \cite{CN} this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci \cite{CSV2} imply the $C^{1+α}$-regularity of the flow and then provide a regularity argument which improves this into $C^{2+α}$-regularity of the flow. The regularity step from $C^{1+α}$ into $C^{2+α}$ does not rely on convexity and can probably be adopted to more general setting.

Convergence of the volume preserving fractional mean curvature flow for convex sets

TL;DR

The paper proves that the volume-preserving fractional mean curvature flow for convex initial sets remains smooth for all time and converges exponentially to a ball. It achieves this by first obtaining uniform -regularity from convexity-based curvature bounds, then deriving a height-function formulation and upgrading regularity to via sphere-adapted parabolic Schauder estimates. The key technical advance is a robust regularity step for the flow that does not rely on convexity for the final stage and leverages a quasilinear formulation of the differentiated flow. Consequently, the flow does not develop singularities and, by existing results, converges exponentially fast to a translated ball, extending Huisken-type convergence to a nonlocal, fractional setting with a solid regularity framework.

Abstract

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga \cite{CN} this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci \cite{CSV2} imply the -regularity of the flow and then provide a regularity argument which improves this into -regularity of the flow. The regularity step from into does not rely on convexity and can probably be adopted to more general setting.
Paper Structure (7 sections, 10 theorems, 234 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 234 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Hölder norms and Schauder estimate
  4. Preliminary results
  5. Equation for the height function
  6. Schauder estimates for the fractional parabolic equation
  7. $C^{2+\alpha}$-estimates for the flow

Key Result

Proposition 2.1

Assume $l_1$ and $l_2$ are positive numbers, $\theta \in (0,1)$ and denote Then there is a constant $C\geq 1$ such that for every smooth function $u : \mathbb{S}^n \to \mathbb{R}$ it holds

Figures (1)

  • Figure 2.1: A picture of the argument used in the proof of Proposition \ref{['prop.curvaturebound']}. In green we have the set $E_k$, in orange $\tilde{E}_k$, i.e. the reflection of $E_k$ over the plane $\{x_{n+1}=0\}$ and in blue the set $U_k$

Theorems & Definitions (20)

  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 10 more