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

TL;DR

This paper establishes curvature and separation estimates for stable solutions of the Allen-Cahn equation on 3-manifolds, confirming conjectures about minimal surfaces' multiplicity, index, and existence in generic metrics, with implications for Yau's conjecture.

Contribution

It proves the multiplicity one and index lower bound conjectures for minimal surfaces arising from Allen-Cahn solutions in 3D, linking PDE analysis with geometric topology.

Findings

Minimal surfaces from Allen-Cahn solutions are two-sided with multiplicity one.

Confirmed Yau's conjecture on infinitely many minimal surfaces in 3-manifolds.

Established Morse index and area estimates for minimal surfaces in generic metrics.

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 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, ..., a two-sided embedded minimal surface with Morse index $p$ and area $p^{1/3}$, as conjectured by Marques-Neves.