An Allen-Cahn Energy on Hypersurfaces and Index of Minimizers

Prof. Jared Marx-Kuo (University of Stanford)
An Allen-Cahn Energy on Hypersurfaces and Index of Minimizers

Abstract 
The Allen-Cahn equation is a well known model for minimal surfaces. We will briefly review the history between the two objects and then define an Allen-Cahn based energy functional on hypersurfaces of a closed manifold. For critical points of this energy, the morse index and nullity agrees with the original Allen-Cahn index and nullity. We use this to compute the index and nullity of all Allen-Cahn solutions on the unit circle. We also discuss the non-locality of the first and second variations of our Allen-Cahn energy, and its connections to the Dirichlet-to-Neumann map for the Allen-Cahn equation.