Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

TL;DR

This paper proves that in manifolds with positive Ricci curvature, minmax solutions from Allen--Cahn energy concentrate on multiplicity-one minimal hypersurfaces, extending known results to higher dimensions and establishing existence of such hypersurfaces.

Contribution

It establishes a new multiplicity-one minimal hypersurface existence result using a geometric minmax approach via Allen--Cahn energy for manifolds with positive Ricci curvature.

Findings

Minmax solutions concentrate on multiplicity-one hypersurfaces

Existence of two-sided closed minimal hypersurfaces in positive Ricci curvature manifolds

Extension of multiplicity-one results to higher dimensions

Abstract

We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by the recent work by Chodosh--Mantoulidis). The argument developed here is geometric in flavour and exploits directly the minmax characterization of the solutions. An immediate corollary is that every compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ and positive Ricci curvature admits a two-sided closed minimal hypersurface, possibly with a singular set of dimension at most $n-7$. This existence result also follows from multiplicity-$1$ results developed within the Almgren--Pitts framework, see works by Ketover-Marques-Neves, Zhou, Marques-Neves, Ramirez-Luna.