Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists uε, a critical point of the Allen-Cahn energy Eε(u) = ε2 ∫ |∇u|2 + ∫(1 − u2)2, whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function uε being a critical point of Eε under some volume constraint.
"From Constant mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions." J. Differential Geom. 64 (3) 359 - 423, July, 2003. https://doi.org/10.4310/jdg/1090426999