Project Euclid

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, The theory of varifolds, and J. Pitts, Existence and regularity of minimal surfaces on Riemannian manifold, 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.