The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus

TL;DR

The paper analyzes the long-time behavior of flat-flow solutions to the planar Mullins-Sekerka flow and the area-preserving curvature flow in the periodic setting $\mathbb{T}^2$. By leveraging the gradient-flow structure, a minimizing-movement discretization, and a new sharp quantitative Alexandrov inequality for periodic sets, it proves exponential convergence of flat flows to explicit configurations: either a finite union of disjoint disks with equal area or a finite union of disjoint strips (or their complements). A key technical advance is the quadratic stability bound linking the curvature deficit $\|\kappa_E-\overline{\kappa_E}\|_{L^2(\partial E)}$ to the proximity to canonical limit configurations, carefully handling non-uniqueness in the strip case. The results extend the understanding of asymptotics from Euclidean to periodic geometries and provide a robust framework for identifying and estimating convergence rates for these geometric flows.

Abstract

We study the asymptotic behavior of flat flow solutions to the periodic and planar two-phase Mullins-Sekerka flow and area-preserving curvature flow. We show that flat flows converge to either a finite union of equally sized disjoint disks or to a finite union of disjoint strips or to the complement of these configurations exponentially fast. A key ingredient in our approach is the derivation of a sharp quantitative Alexandrov inequality for periodic smooth sets.

Theorems & Definitions (25)

• Theorem
• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof