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

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