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Analysis Seminar

Thursday, November 16, 2017 - 3:00pm

Fritz Hiesmayr

University of Cambridge


University of Pennsylvania


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.