  We study the asymptotic behavior of Dirichlet minimizers to the Allen--Cahn
equation on manifolds with boundary, and we relate the Neumann data to the
geometry of the boundary. We show that Dirichlet minimizers are asymptotically
local in orders of $\epsilon$ and compute expansions of the solution to high
order. A key tool is showing that the linearized allen-cahn operator is
invertible at the heteroclinic solution, on functions with $0$ boundary
condition. We apply our results to separating hypersurfaces in closed
Riemannian manifolds. This gives a projection theorem about Allen--Cahn
solutions near minimal surfaces, as constructed by Pacard--Ritore.
