Phase transitions and minimal hypersurfaces in hyperbolic space]

TL;DR

This paper explores the existence and properties of minimal hypersurfaces in hyperbolic space using phase transition models, variational methods, and $$-convergence, linking geometric measure theory with PDE techniques.

Contribution

It introduces a novel approach combining Cahn-Hilliard approximation with blow-up and comparison principles to establish existence results for minimal hypersurfaces with prescribed boundary behavior.

Findings

Existence of entire local minimizers with prescribed behavior at infinity.

Limit analysis via $$-convergence yields minimal hypersurfaces with boundary at infinity.

Connections established with previous geometric measure theory results.

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 behaviour at infinity. Then, we study the limit as the length scale tends to zero through a $\Gamma$-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in M. Anderson and U. Lang using geometric measure theory.