Stability of the ball under volume preserving fractional mean curvature flow

TL;DR

This paper studies the behavior of nearly spherical sets under volume-preserving fractional mean curvature flow, proving long-term stability and convergence to a ball, with implications for convex shapes and periodic graphs.

Contribution

It establishes long-time existence and asymptotic stability results for fractional mean curvature flow, extending understanding of geometric evolution under volume constraints.

Findings

Nearly spherical sets converge to a ball over time.

Convex initial data also stabilize to a spherical shape.

Exponential convergence shown for periodic graph flows.

Abstract

We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.