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 [39] 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; [29,35].

We study global variational properties of the space of solutions to −ε 2 ∆u + W ′ (u) = 0 on any closed Riemannian manifold M . Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if ε is not small enough, in terms of the Cheeger constant of M , then there are no interesting solutions. However, we show that the number of minmax solutions to the equation above goes to infinity as ε → 0 and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed ε, as shown recently by G. Smith. We also show that the energy of the min-max solutions accumulate, as ε → 0, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with RicM > 0, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet-Rosenberg. Finally, we prove that the min-max energy values are bounded from below by the widths of the area functional as defined by Marques-Neves.Both authors were partly supported by CNPq-Brazil and NSF-DMS-1104592. 1 c ε (p) ≤ Cp 1 n for all p ∈ N.The proofs of these sublinear bounds are inspired by similar computations by Gromov [28], Guth [30] and Marques-Neves [40] for sweepouts of the area functional. In this sense, our work brings the analogy between phase transitions and minimal hypersurfaces to a high parameter global variational context.Combining Theorem 3 with the upper bound from Theorem 4, we obtainCorollary 5. On any closed Riemanian manifold the number of solutions to the elliptic Allen-Cahn equation (1) goes to infinity as ε → 0. Moreover, for every p, Theorem A implies that the min-max solutions at the level c ε (p) have a limit-interface that is a smooth embedded minimal hypersurface.This article ends in Section 6, where we compare the sequence of values lim inf ε→0 c ε (p) with the widths ω p (M ) of the area functional on M , as defined by Marques-Neves [40]. These widths form a sequence of real values which Gromov [28] proposed to consider as a nonlinear spectrum of M , by analogy

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We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich-Marques-Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song, respectively. We also prove the density of separating limit interfaces for generic metrics in dimension 3, based on the recent work of Chodosh-Mantoulidis, and for generic metrics on manifolds containing only separating minimal hypersurfaces, e.g. Hn(M, Z2) = 0, for 4 ≤ n + 1 ≤ 7. These provide alternative proofs of Yau's conjecture on the existence of infinitely many minimal hypersurfaces for generic metrics on each setting, using the Allen-Cahn approach.

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