Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds II, Brakke's flows

TL;DR

This paper proves that solutions to the Allen-Cahn equation in space forms converge to Brakke's mean curvature flow, extending previous Euclidean results to Riemannian manifolds and utilizing a local monotonicity formula for analysis.

Contribution

It generalizes convergence results of Allen-Cahn solutions to Brakke's flow from Euclidean space to Riemannian manifolds, introducing new techniques for measure convergence.

Findings

Convergence of Allen-Cahn solutions to Brakke's flow in space forms.

Establishment of density bounds for limiting measures.

Extension of previous Euclidean results to Riemannian manifolds.

Abstract

We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen-Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [22], to get various density bounds for the limiting measures.