Min-max for phase transitions and the existence of embedded minimal hypersurfaces

By Marco A. M. Guaraco. 2015

135 citations recorded by the source corpus.

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren-Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min-max theories.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/82ab4335022cb61f4369b4428febeb1b28483911

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “[14] M. A. M. Guaraco.”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “D G ] 1 4 M ay 2 01 8 Allen–Cahn min-max constructions were recently carried out by Guaraco [Gua18] and Gaspar–Guaraco [GG16].”
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].”
The Allen–Cahn equation on closed manifolds (2016, methodology): “In [29], based on the recent works [33,63], the following theorem was proved.”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “We refer to [39, 6, 10, 17, 9, 23, 7, 47, 50, 32] for other variants of this theory.”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “101, 42 pp. [16] Gaspar, P., Guaraco, M.”