Index and spectrum of minimal hypersurfaces arising from the Allen-Cahn construction

Fritz Hiesmayr, Cambridge University
Fine Hall 110

The Allen-Cahn construction is a method for constructing minimal surfaces of codimension 1 in closed manifolds. In this approach, minimal hypersurfaces arise as the weak limits of level sets of critical points of the Allen-Cahn energy functional. This talk will relate the variational properties of the Allen-Cahn energy to those of the area functional on the surface arising in the limit, under the assumption that the limit surface has a unit normal section. In this case, bounds for the Morse indices of the critical points lead to a bound for the Morse index of the limit minimal surface. As a corollary, minimal hypersurfaces arising from an Allen-Cahn p-parameter min-max construction have index at most p. An analogous argument also establishes a lower bound for the spectrum of the Jacobi operator of the limit surface.