Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen--Cahn

TL;DR

This paper develops a modified minmax method using Allen--Cahn approximation to prove the existence of smooth, multiplicity-one minimal hypersurfaces in compact Riemannian manifolds of dimension 3 to 7 with bumpy metrics.

Contribution

It introduces a new minmax construction allowing optimization over valley points, ensuring the resulting minimal hypersurface has multiplicity one in the specified dimensions.

Findings

Existence of multiplicity-one minimal hypersurfaces in dimensions 3 to 7.

Every compact manifold with a bumpy metric admits a smooth, two-sided minimal hypersurface.

The method generalizes previous approaches and confirms the generic existence of such hypersurfaces.

Abstract

In Guaraco's 2018 work a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold $N^{n+1}$ with $n\geq 2$. This was achieved by employing an Allen--Cahn approximation scheme and a one-parameter minmax for the Allen--Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to $0$). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we consider a minmax construction that is a modification of the one in the aforementioned work, by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We prove that, when $2\leq n\leq 6$ and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity $1$. (When $n=2$ this conclusion also follows from Chodosh--Mantoulidis's recent work.) As immediate corollary we obtain that every compact Riemannian manifold of dimension $n+1$, $2\leq n\leq 6$, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou's recent result for minmax constructions via Almgren--Pitts theory).