Spectrum and index of two-sided Allen-Cahn minimal hypersurfaces

The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension $n + 1 \geq 3$ contains a minimal hypersurface with a singular set of Hausdorff dimension at most $n-7$. This proof avoids the Almgren--Pitts geometric min-max procedure for the area functional that was instrumental in the original proof, and is instead based on a considerably simpler PDE min-max construction of critical points of the Allen--Cahn functional. Here we prove a spectral lower bound for the hypersurfaces arising from this construction. This directly implies an upper bound for the Morse index of the hypersurface in terms of the indices of the critical points, provided it is two-sided. In particular, two-sided hypersurfaces arising from Guaraco's construction have Morse index at most $1$. Finally, we point out by an elementary inductive argument how the regularity of the hypersurface follows from the corresponding result in the stable case.