Geometric Variations of an Allen-Cahn Type Energy

Abstract

In this thesis (adapted from joint work with Jared Marx-Kuo) a new Allen–Cahn type functional,BEε, is introduced, which defines an energy on separating hypersurfaces, Y , of closed Riemannian Manifolds. In Chapter 2 we establish Γ-convergence of BEε to the area functional as ε → 0. In Chapter 3 we compute first and second variations of this functional under hypersurface pertru-bations. We then compute an explicit expansion for the variational formula as ε → 0. A key com- ponent of this proof is the invertibility of the linearized Allen–Cahn equation about a solution, on the space of functions vanishing on Y .In Chapter 4 we also study the index and nullity of BEε and relate it to the usual Allen–Cahn index of a corresponding solution vanishing on Y . We apply the second variation formula and index theorems to show that the family of 2p-dihedrally symmetric solutions to Allen–Cahn on S 1 have index 2p − 1 and nullity 1.