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 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. © 2003 Applied Probability Trust.