Constant (nonlocal) Mean curvature surfaces

Abstract

The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. Alike their local counterpart, 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 dibock-copolymer and the study of overdetermined boundary value problems.