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 $\sim p^{\frac{1}{3}}$, as conjectured by Marques--Neves.
Citation
Otis Chodosh. Christos Mantoulidis. "Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates." Ann. of Math. (2) 191 (1) 213 - 328, January 2020. https://doi.org/10.4007/annals.2020.191.1.4
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