## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 64, Number 3 (2003), 359-423.

Dates
First available in Project Euclid: 21 July 2004

https://projecteuclid.org/euclid.jdg/1090426999

Digital Object Identifier
doi:10.4310/jdg/1090426999

Mathematical Reviews number (MathSciNet)
MR2032110

Zentralblatt MATH identifier
1070.58014

#### Citation

Pacard, Frank; Ritoré, Manuel. From Constant mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions. J. Differential Geom. 64 (2003), no. 3, 359--423. doi:10.4310/jdg/1090426999. https://projecteuclid.org/euclid.jdg/1090426999