On the evolution by fractional mean curvature

TL;DR

This paper investigates the evolution of smooth solutions under fractional mean curvature flow, establishing fundamental principles like comparison, uniqueness, and finite extinction, along with evolution equations for fractional geometric quantities.

Contribution

It introduces new comparison and evolution principles for fractional mean curvature flow, ensuring properties like preservation of fractional curvature and smoothness of solutions.

Findings

Comparison principle and uniqueness for fractional mean curvature flow

Finite extinction time for compact solutions

Preservation of positive fractional curvature

Abstract

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain quantities (such as positive fractional curvature) and smoothness of graphical evolutions.