Constant Nonlocal Mean Curvatures surfaces and related problems

TL;DR

This paper explores the properties of nonlocal mean curvature, a geometric quantity extending classical mean curvature, and surveys recent results on surfaces with constant NMC and related overdetermined boundary problems.

Contribution

It provides a detailed analysis of the properties of nonlocal mean curvature and reviews recent advances in understanding surfaces with constant NMC and their connection to boundary value problems.

Findings

Properties of nonlocal mean curvature described

Surveys of recent results on constant NMC surfaces

Connections to overdetermined boundary problems elucidated

Abstract

The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear differential operators that are involved when it acts on graphs. We also survey recent results on surfaces having constant NMC and describe their intimate link with some problems arising in the study of overdetermined boundary value problems.