A Gluing Construction for Prescribed Mean Curvature

TL;DR

This paper introduces a gluing method to construct hypersurfaces with nearly constant prescribed mean curvature by perturbing unions of spheres aligned along a line segment, under specific integral conditions.

Contribution

It develops a new gluing construction technique for hypersurfaces with prescribed mean curvature, extending previous methods to configurations of multiple spheres.

Findings

Constructed hypersurfaces with approximately constant prescribed mean curvature.

Identified conditions involving integral moments for the existence of these hypersurfaces.

Demonstrated the perturbation approach for assembling spheres into desired shapes.

Abstract

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respect the prescribed mean curvature function.