Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

TL;DR

This paper studies the long-term behavior of a discrete approximation to the volume-preserving fractional mean curvature flow, showing exponential convergence to a ball and establishing a fractional Alexandrov estimate.

Contribution

It introduces a quantitative Alexandrov theorem in the fractional setting and proves exponential convergence of the discrete flow to a sphere.

Findings

Discrete flow converges exponentially to a sphere

Quantitative Alexandrov estimate established for fractional mean curvature

Existence of flat flows as limits of discrete approximations

Abstract

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.