Asymptotics and geometric flows for a class of nonlocal curvatures

TL;DR

This paper studies a broad class of nonlocal curvatures, analyzing their asymptotic behavior and geometric flows, and connecting them to known curvature variants like fractional and anisotropic fractional curvature.

Contribution

It introduces a unified framework for nonlocal curvatures, examines their limits, and establishes existence and stability of solutions for related geometric evolution equations.

Findings

Unified framework for various nonlocal curvatures

Asymptotic limits recover known curvature results

Proved existence and stability of viscosity solutions

Abstract

We consider a family of nonlocal curvatures determined through a kernel which is symmetric and bounded from above by a radial and radially non-increasing profile satisfying an integrability condition. It turns out that such definition encompasses various variants of nonlocal curvatures that have already appeared in the literature, including fractional curvature and anisotropic fractional curvature. The main task undertaken in the article is to study the limit behaviour of the introduced nonlocal curvatures under an appropriate limiting procedure. This enables us to recover known asymptotic results e.g. for the fractional curvature and for the anisotropic fractional curvature. For the convergence of anisotropic fractional curvatures we identify the limit object as the nonlocal curvature being the first variation of the related anisotropic fractional perimeter. We also prove existence, uniqueness and stability of viscosity solutions to the corresponding level-set parabolic Cauchy problem formulated in terms of the investigated nonlocal curvature.