Phase transitions and minimal hypersurfaces in hyperbolic space

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, 21] using geometric measure theory. © Taylor & Francis Group, LLC.