From Constant mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions

Abstract

Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists u_ε, a critical point of the Allen-Cahn energy E_ε(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 u_ε being a critical point of E_ε under some volume constraint.