[Paper Review] Singularities of Mean Curvature Flow of Surfaces with Additional Forces

Ao Sun|arXiv (Cornell University)|Aug 12, 2018

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper establishes that the blow-up limit of a mean curvature flow of smoothly embedded surfaces in ℝ³ with additional forces—under bounded $L^\infty$ force fields and finite entropy—converges to a smoothly embedded self-shrinker. The proof extends Ilmanen's regularity result for standard mean curvature flow by controlling non-monotonic geometric quantities and showing the influence of external forces vanishes under parabolic rescaling, ensuring the limit satisfies the self-shrinker equation $\vec{H} + \frac{x^\perp}{-2t} = 0$. This result confirms the smoothness of singularity models even with external forces, under topological and entropy constraints.

ABSTRACT

In this paper we study the blow up sequence of mean curvature flow of surfaces in $\mathbb R^3$ with additional forces. We prove that the blow up limit of a mean curvature flow of smoothly embedded surfaces with additional forces with finite entropy is a smoothly embedded self-shrinker.

Motivation & Objective

• To extend Ilmanen's 1995 regularity result for mean curvature flow to the case with additional forces.
• To establish that the blow-up limit of a mean curvature flow with bounded $L^\infty$ external forces is a smoothly embedded self-shrinker.
• To address the challenge that geometric quantities (e.g., area, curvature) are no longer monotone under additional forces.
• To show the influence of the external force becomes negligible in the blow-up limit, preserving smoothness of the limit surface.
• To generalize the regularity theory of singularity models to flows with external forces, under topological and entropy constraints.

Proposed method

• Uses parabolic rescaling (blow-up sequence) $M_t^{\lambda_i} := \lambda_i^{-1}(M_{s + \lambda_i^2 t} - y)$ to analyze singularities at $(y,s)$.
• Applies Huisken’s monotonicity formula and Brakke’s weak flow framework to establish weak convergence to a self-similar Brakke flow.
• Employs Simon’s graph decomposition theorem to control local geometry and prove regularity away from concentration points.
• Uses Allard’s regularity theorem to show that low-area and low-curvature regions are graphs of $C^{\infty}$ functions, implying smoothness.
• Applies the maximum principle and a result from Colding-Silberman (1985) to extend the smooth structure across concentration points.
• Controls non-monotonic geometric quantities via a comparison lemma (Lemma 6.1) to bound growth under the influence of external forces.

Experimental results

Research questions

• RQ1 Does the blow-up limit of a mean curvature flow with additional forces remain smoothly embedded under finite entropy and bounded force fields?
• RQ2 Can the regularity result for standard mean curvature flow be extended to flows with external forces?
• RQ3 How do additional forces affect the monotonicity of geometric quantities like area and curvature in the blow-up process?
• RQ4 What conditions ensure that the influence of external forces vanishes in the blow-up limit?
• RQ5 Is the self-shrinker equation $\vec{H} + \frac{x^\perp}{-2t} = 0$ preserved in the limit when forces are present?

Key findings

• The blow-up limit of a mean curvature flow with additional forces and finite entropy is a smoothly embedded self-shrinker.
• The support of the blow-up limit satisfies the self-shrinker equation $\vec{H} + \frac{x^\perp}{-2t} = 0$ for $t < 0$, even with external forces.
• The set of concentration points $Q$ is discrete and isolated, and the limit is smoothly embedded away from $Q$.
• The influence of the bounded $L^\infty$ force field becomes negligible in the blow-up limit, allowing the use of standard regularity tools.
• The smooth structure extends across concentration points via a minimal surface argument in the Gaussian metric, preserving embeddedness.
• The result holds even if the force field is bounded only on the region occupied by the flow, not globally.