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

2015/05/25 by Marco A. M. Guaraco, Guaraco, Marco A. M. · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1505.06698

openalex publication_date 2015/05/25 · openalex created_date 2022/09/22 · openalex updated_date 2026/07/28

Abstract

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

Cited by

• Anisotropic min-max via phase transitions
• Varifold convergence of free boundary Allen--Cahn equation

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