Min–max hypersurface in manifold of positive Ricci curvature

Abstract

In this paper, we study the shape of the min–max minimal hypersurface produced by Almgren–Pitts–Schoen–Simon in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min–max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min–max hypersurface. In particular, we show that the min–max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface.

As an essential technical tool, we prove a stronger version of the discretization theorem. The discretization theorem, first developed by Marques–Neves in their proof of the Willmore conjecture, is a bridge to connect sweepouts appearing naturally in geometry to sweepouts used in the min–max theory. Our result removes a critical assumption of Marques–Neves in their proof, called the no-mass concentration condition, and hence confirms a conjecture by Marques–Neves in their proof.