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