MIN-MAX MINIMAL HYPERSURFACE

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in [AF2, P] corresponding to the fundamental class of a Riemannian manifold (Mn+1, g) of positive Ricci curvature with 2 <= n <= 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.