Orientability of min-max hypersurfaces in manifolds of positive Ricci curvature

TL;DR

This paper proves that in positively Ricci curved manifolds, the minimal hypersurface realizing the Almgren-Pitts width is orientable, index 1, and multiplicity 1, extending previous results to higher dimensions.

Contribution

It extends the orientability and regularity results of minimal hypersurfaces to dimensions greater than or equal to 8 in manifolds with positive Ricci curvature.

Findings

The Almgren-Pitts width is achieved by an orientable index 1 minimal hypersurface.

The minimal hypersurface has multiplicity 1 and optimal regularity.

Results extend previous work to higher dimensions (n+1 ≥ 8).

Abstract

Let $M^{n+1}$ be an orientable compact Riemannian manifold with positive Ricci curvature. We prove that the Almgren-Pitts width of $M^{n+1}$ is achieved by an orientable index $1$ minimal hypersurface with multiplicity $1$ and optimal regularity. This extends to dimensions $n+1\geq 8$ the results of Ketover-Marques-Neves arXiv:1601.04514v1 [math.DG] and X. Zhou arXiv:1504.00966v2 [math.DG].