Min-max minimal hypersurface in manifolds with convex boundary and $Ric\geq 0$

TL;DR

This paper proves that in certain manifolds with non-negative Ricci curvature and convex boundary, the min-max minimal hypersurface is always orientable, of index one, and appears with multiplicity one, advancing understanding of minimal hypersurfaces in geometric analysis.

Contribution

It establishes the orientability, index, and multiplicity properties of min-max minimal hypersurfaces in manifolds with convex boundary and non-negative Ricci curvature.

Findings

Min-max minimal hypersurfaces are orientable.

They have index one.

They occur with multiplicity one.

Abstract

Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.