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 is two-sided.

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.

### Analysis Seminar

Thursday, November 16, 2017 - 3:00pm

#### Fritz Hiesmayr

University of Cambridge