In light of the connection between the theory of phase transitions and the critical points for the area functional we describe a variational construction of solutions to the elliptic Allen-Cahn equation in closed manifolds and the corresponding limit interfaces, that is the minimal hypersurfaces which arise as weak limits of the level sets of such solutions. This construction is inspired by min-max theory of minimal hypersurfaces and it yields a sequence of critical values for the Allen-Cahn energy functional comparable to the p-widths of the ambient manifold, as defined by Gromov and studied by Guth and by Marques-Neves. This is joint work with M. Guaraco.

We also describe an upper bound for the stability index of limit interfaces which arise from solutions with uniformly bounded energy and index in terms of the Morse index of these solutions obtained by comparing the second inner variation of the energy functional to the second variation of the area.

### Geometry-Topology Seminar

Thursday, January 25, 2018 - 4:30pm

#### Pedro Gaspar

Princeton University