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

2020/04/21 by Bellettini, Costante
#53A10 49Q20 35J20 35J61 35K58 49Q05 53C21 58J60 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2004.10112

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 Nⁿ⁺¹ with n≥ 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 ≤ n-7. This result is new for n≥ 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 Nⁿ⁺¹ with n≥ 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.

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