Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
By Otis Chodosh, Christos Mantoulidis. 2018
182 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Materials Chemistry
View Paper: https://www.semanticscholar.org/paper/b5523c6f71069b99bb7d809761d0585aa42cfa1b
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In Applied Mathematics
In Materials Chemistry
Citations in the corpus
- Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “[7] O. Chodosh and C. Mantoulidis.”
- On the Multiplicity One Conjecture in min-max theory (2019, unknown): “Result (b) was also obtained by Chodosh-Mantoulidis [5] in dimension three (n+ 1) = 3.”
- Existence of hypersurfaces with prescribed mean curvature I – generic min-max (2018, background): “Consequently, the min-max program has seen a series of developments in various directions, including (but not limited to) [16, 28, 37, 44, 42, 15, 41, 36, 12, 46, 57].”