Existence of minimal hypersurfaces in complete manifolds of finite volume

TL;DR

This paper proves that complete non-compact manifolds with finite volume always contain minimal hypersurfaces of finite volume, using a novel sweeping technique to construct disjoint hypersurfaces with controlled volume.

Contribution

It introduces a new method for constructing minimal hypersurfaces in finite volume manifolds by refining sweepouts to disjoint hypersurfaces with similar volume bounds.

Findings

Existence of minimal hypersurfaces in all complete finite volume manifolds.

A new sweeping technique for hypersurfaces with volume control.

Construction of disjoint hypersurfaces approximating sweepouts.

Abstract

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region $U$ can be swept out by a family of hypersurfaces of volume at most $V$, then it can be swept out by a family of mutually disjoint hypersurfaces of volume at most $V + \varepsilon$.