Combinatorics

Electrical engineering

Pure mathematics

Allen–Cahn equation

Mathematics

Manifold (fluid mechanics)

World Wide Web

Engineering

Mathematical analysis

Morse code

Multiplicity (mathematics)

Morse theory

Curvature

Geometry

Index (typography)

Computer science

Conjecture

Minimal surface

Bounded function

Upper and lower bounds

Mechanical engineering

Annals of Mathematics

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.

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