7 mar 2024 -- 18:30   [open in google calendar]

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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.