Minimal graphs in H^n xR and R^{n+1}

TL;DR

This paper constructs geometric barriers and proves existence and uniqueness results for minimal graphs in hyperbolic space cross real line and Euclidean space, including solutions with infinite boundary data and Scherk-type hypersurfaces.

Contribution

It introduces new methods for solving the Dirichlet problem for minimal graphs in hyperbolic and Euclidean spaces with complex boundary conditions.

Findings

Existence and uniqueness of solutions in convex polyhedra in H^n x R.

Construction of Scherk-type minimal hypersurfaces with alternating infinite boundary data.

Extension of results to Euclidean space R^{n+1}.

Abstract

We construct geometric barriers for minimal graphs in H^n xR. We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in H^n extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on the other faces. In H^n xR, we solve the Dirichlet problem for the vertical minimal equation in a C^0 convex domain taking arbitrarily continuous finite boundary and asymptotic boundary data. We prove the existence of another Scherk type hypersurface, given by the solution of the vertical minimal equation in the interior of certain admissible polyhedron taking alternatively infinite values +\infty and -\infty on adjacent faces of this polyhedron. Those polyhedra may be chosen convex or non convex. We establish analogous results for minimal graphs when the ambient is the Euclidean space R^ {n+1}.