Alexandrov Theorem for nonlocal curvature

TL;DR

This paper extends Alexandrov's classical theorem to a nonlocal setting, showing that smooth sets with constant nonlocal mean curvature must be Euclidean balls, under a broad class of kernels.

Contribution

It introduces a general nonlocal mean curvature framework with a verifiable condition ensuring the theorem's validity, unifying various existing nonlocal curvature models.

Findings

Theorem establishing nonlocal Alexandrov result under new conditions

Derived a formula for the tangential derivative of nonlocal mean curvature

Applied the method of moving planes to prove the main result

Abstract

In this article we obtain a nonlocal version of the Alexandrov Theorem which asserts that the only set with sufficiently smooth boundary and of constant nonlocal mean curvature is an Euclidean ball. We consider a general nonlocal mean curvature given by a radial and monotone kernel and we formulate an easy-to-check condition which is necessary and sufficient for the nonlocal version of the Alexandrov Theorem to hold in the treated context. Our definition encompasses numerous examples of various nonlocal mean curvatures that have been already studied in the literature. To prove the main result we obtain a specific formula for the tangential derivative of the nonlocal mean curvature and combine it with an application of the method of moving planes.