Minimal hypersurfaces of least area

TL;DR

This paper investigates the existence and properties of minimal hypersurfaces with least area in certain Riemannian manifolds, establishing their existence via minimization or min-max methods and deriving area bounds in hyperbolic 3-manifolds.

Contribution

It proves the existence of least area minimal hypersurfaces with index at most one in dimensions 2 to 6, using both minimization and min-max techniques, and applies results to hyperbolic 3-manifolds.

Findings

Existence of least area minimal hypersurfaces with index ≤ 1

Construction via minimization and min-max methods

Lower area bounds in hyperbolic 3-manifolds

Abstract

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max methods: they have index at most $1$. We apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic $3$-manifolds.