Minimal hypersurfaces of least area

Abstract

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold $(2 \leq n \leq 6)$ 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.