Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

Abstract

The Allen–Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang–Wei) of the Allen–Cahn equation on a $3$-manifold. Using these, we are able to show that for generic metrics on a $3$-manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity one-conjecture and the index lower bound conjecture of Marques–Neves in $3$-dimensions regarding min-max constructions of minimal surfaces.

Allen–Cahn min-max constructions were recently carried out by Guaraco and Gaspar–Guaraco. Our resolution of the multiplicity-one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau’s conjecture on infinitely many minimal surfaces in a $3$-manifold with a generic metric (recently proven by Irie–Marques–Neves) with new geometric conclusions. Namely, we prove that a $3$-manifold with a generic metric contains, for every $p = 1, 2, 3, \ldots $, a two-sided embedded minimal surface with Morse index $p$ and area $\sim p^{\frac 13}$, as conjectured by Marques–Neves.