Min-max hypersurface in manifold of positive Ricci curvature

TL;DR

This paper investigates the properties of min-max minimal hypersurfaces in positively Ricci curved manifolds, characterizing their singularities, Morse index, and multiplicity, and advances the discretization technique in min-max theory.

Contribution

It introduces a stronger discretization theorem that removes a key assumption, confirming a conjecture by Marques-Neves and extending min-max hypersurface analysis to all dimensions.

Findings

Characterized the Morse index, area, and multiplicity of min-max hypersurfaces.

Proved the min-max hypersurface is either orientable with index one or a double cover of a non-orientable stable hypersurface.

Established a stronger discretization theorem removing the no mass concentration condition.

Abstract

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} 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 \cite{MN12}, 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 \cite{MN12}, called the no mass concentration condition, and hence confirms a conjecture by Marques-Neves in \cite{MN12}.