Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds

TL;DR

This paper develops a min-max theory for constructing prescribed mean curvature hypersurfaces in noncompact manifolds, leading to new existence results for such hypersurfaces in Euclidean space and finite volume manifolds.

Contribution

It introduces a novel min-max framework for prescribed mean curvature hypersurfaces in noncompact settings, applicable to sign-changing functions outside compact sets.

Findings

Existence of closed prescribed mean curvature hypersurfaces in Euclidean space.

Existence of complete finite area constant mean curvature hypersurfaces in finite volume manifolds.

New techniques for noncompact geometric variational problems.

Abstract

We develop a min-max theory for hypersurfaces of prescribed mean curvature in noncompact manifolds, applicable to prescription functions that do not change sign outside a compact set. We use this theory to prove new existence results for closed prescribed mean curvature hypersurfaces in Euclidean space and complete finite area constant mean curvature hypersurfaces in finite volume manifolds.