Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\leq n\leq 6$

TL;DR

This paper characterizes the shape, index, area, and multiplicity of min-max minimal hypersurfaces in positively Ricci curved manifolds, revealing they are either orientable with index one or double covers of non-orientable minimal hypersurfaces.

Contribution

It provides a detailed description of the min-max hypersurface structure in manifolds with positive Ricci curvature for dimensions 2 to 6, including their orientability and index.

Findings

Min-max hypersurface is either orientable with index one or a double cover of a non-orientable hypersurface.

The hypersurface has least area among all closed embedded minimal hypersurfaces.

Characterization applies to manifolds with positive Ricci curvature and dimensions 2 to 6.

Abstract

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \cite{A2}\cite{P} corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with $2\leq n\leq 6$. We characterize the Morse index, area and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.