Geometric Variations of an Allen-Cahn Energy on Hypersurfaces

We introduce an Allen-Cahn type functional, ${\text{BE}}_{\epsilon }$, that defines an energy on separating hypersurfaces, $Y$, of closed Riemannian Manifolds. We establish $\mathrm{\Gamma }$-convergence of ${\text{BE}}_{\epsilon }$ to the area functional, and compute first and second variations of this functional under hypersurface pertrubations. We then compute an explicit expansion for the variational formula as $\epsilon \to 0$. A key component of this proof is the invertibility of the linearized Allen-Cahn equation about a solution, on the space of functions vanishing on $Y$. We also relate the index and nullity of ${\text{BE}}_{\epsilon }$ to the Allen-Cahn index and nullity 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 $1$.