Phase Transitions and Minimal Hypersurfaces in Hyperbolic Space

Abstract

The purpose of this paper is to investigate the Cahn–Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behavior at infinity. Then, we study the limit as the length scale tends to zero through a Γ-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in [3 Anderson , M. ( 1982 ) Complete minimal varieties in hyperbolic space . Invent. Math. 69 : 477 – 494 .[Crossref], [Web of Science ®] , [Google Scholar], 21 Lang , U. ( 1995 ). The existence of complete minimizing hypersurfaces in hyperbolic manifolds . Internat. J. Math. 6 : 45 – 58 . [Google Scholar]] using geometric measure theory.