Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

TL;DR

This paper proves convergence of a multiphase, volume-preserving mean curvature flow scheme on weighted manifolds, extending the classical thresholding method to a data science context with a new weak solution concept.

Contribution

It introduces a convergence proof for a multiphase, volume-preserving mean curvature flow scheme on weighted manifolds, with a new De Giorgi solution concept and energy analysis.

Findings

Convergence of the scheme to the weighted, volume-preserving mean curvature flow.

Development of estimates for heat kernel derivatives to facilitate limit passage.

Introduction of a new weak solution concept based on energy dissipation.

Abstract

The famous thresholding scheme by Merriman, Bence, and Osher (Motion of multiple junctions: A level set approach. Journal of Computational Physics 112.2 (1994): 334-363.) proved itself as a very efficient time discretization of mean curvature flow. The present paper studies a multiphase, volume constrained version on a weighted manifold that naturally arises in the context of data science. The main result of this work proves convergence to multiphase, weighted, volume-preserving mean curvature flow on a smooth, closed manifold. The proof is only conditional in the sense that convergence of the approximating energy to the weighted perimeter is assumed. These type of assumptions are natural and common in the literature. The proof shows convergence in the energy dissipation inequality, induced by the underlying gradient flow structure, similar to the work of Laux and Otto (The thresholding scheme for mean curvature flow and De Giorgi's ideas for minimizing movements. The Role of Metrics in the Theory of Partial Differential Equations 85 (2020): 63-94.). This leads to a new De Giorgi solution concept that describes weakly the limiting flow. Estimates regarding the derivatives of the heat kernel are developed in order to pass to the limit in the variations of the energy and metric.