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

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In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold N^{n+1} with nge 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 modify the minmax construction of Guaraco (J. Differential Geom. 108(1):91–133, 2018), 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 then prove that, when 2le nle 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 and Mantoulidis (Ann. Math. 191(1):213–328, 2020).) As immediate corollary we obtain that every compact Riemannian manifold of dimension n+1, 2le nle 6, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou X (Ann. Math. (2), 192(3):767–820, 2020) for minmax constructions via Almgren–Pitts theory).