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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 minimalhypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq2$ (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 solutionsconcentrate around a multiplicity-$1$ hypersurface, that may have a singularset of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it isalso implied by the recent work by Chodosh--Mantoulidis). The argumentdeveloped here is geometric in flavour and exploits directly the minmaxcharacterization of the solutions. An immediate corollary is that every compactRiemannian manifold $N^{n+1}$ with $n\geq 2$ and positive Ricci curvatureadmits a two-sided closed minimal hypersurface, possibly with a singular set ofdimension at most $n-7$. This existence result also follows frommultiplicity-$1$ results developed within the Almgren--Pitts framework, seeworks by Ketover-Marques-Neves, Zhou, Marques-Neves, Ramirez-Luna.

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