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July, 2003 From Constant mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions
Frank Pacard, Manuel Ritoré
J. Differential Geom. 64(3): 359-423 (July, 2003). DOI: 10.4310/jdg/1090426999

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.

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Frank Pacard. Manuel Ritoré. "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

Information

Published: July, 2003
First available in Project Euclid: 21 July 2004

zbMATH: 1070.58014
MathSciNet: MR2032110
Digital Object Identifier: 10.4310/jdg/1090426999

Rights: Copyright © 2003 Lehigh University

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Vol.64 • No. 3 • July, 2003
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