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

• 成果类型: Article
• 署名作者: Chodosh, Otis; Mantoulidis, Christos
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.191.1.4
• 发表日期: 2020
• 页码: 213-328
• 关键词: min-max; MORSE INDEX; singular perturbation; phase-transitions; gradient theory; weyl law; hypersurfaces; SPACE; compactness; conjecture

摘要

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 Yaus 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 similar to p(1/3), as conjectured by Marques-Neves.