 Award ID(s):
 1811293
 NSFPAR ID:
 10169770
 Date Published:
 Journal Name:
 Cambridge journal of mathematics
 Volume:
 8
 Issue:
 2
 ISSN:
 21680930
 Page Range / eLocation ID:
 311 – 362
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 AllenCahn equation on a 3manifold. Using these, we are able to show that for generic metrics on a 3manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are twosided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity oneconjecture and the index lower bound conjecture of Marques–Neves in 3dimensions regarding minmax constructions of minimal surfaces. Allen–Cahn minmax constructions were recently carried out by Guaraco and Gaspar–Guaraco. Our resolution of the multiplicityone 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 3manifold with a generic metric (recently proven by Irie–Marques–Neves) with new geometric conclusions. Namely, we prove that a 3manifold with a generic metric contains, for every p = 1, 2, 3,…, a twosided embedded minimal surface with Morse index p and area ~ p13, as conjectured by MarquesNeves.more » « less

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