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

Abstract

AbstractWe address the one‐parameter minmax construction for the 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 with (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci curvature of N is positive then the minmax Allen–Cahn solutions concentrate around a multiplicity‐1 minimal hypersurface (possibly having a singular set of dimension ). This multiplicity result is new for (for it is also implied by the recent work by Chodosh–Mantoulidis). We exploit directly the minmax characterization of the solutions and the analytic simplicity of semilinear (elliptic and parabolic) theory in . While geometric in flavour, our argument takes advantage of the flexibility afforded by the analytic Allen–Cahn framework, where hypersurfaces are replaced by diffused interfaces; more precisely, they are replaced by sufficiently regular functions (from N to ), whose weighted level sets give rise to diffused interfaces. We capitalise on the fact that (unlike a hypersurface) a function can be deformed both in the domain N (deforming the level sets) and in the target (varying the values). We induce different geometric effects on the diffused interface by using these two types of deformations; this enables us to implement in a continuous way certain operations, whose analogues on a hypersurface would be discontinuous. An immediate corollary of the multiplicity‐1 conclusion is that every compact Riemannian manifold with and positive Ricci curvature admits a two‐sided closed minimal hypersurface, possibly with a singular set of dimension at most . (This geometric corollary also follows from results obtained by different ideas in an Almgren–Pitts minmax framework.)