A note on minimal graphs over certain unbounded domains of Hadamard manifolds

TL;DR

This paper investigates the existence of minimal graphs over unbounded domains in Hadamard manifolds with curvature ≤ -1, establishing conditions under which the Dirichlet problem is solvable and necessary convexity conditions.

Contribution

It extends solvability results for minimal graphs to unbounded domains in Hadamard manifolds, identifying convexity and mean convexity conditions as key factors.

Findings

Solvability of the Dirichlet problem under curvature and convexity conditions

Mean convexity of the boundary is necessary for existence

Extension of bounded domain results to unbounded domains

Abstract

Given an unbounded domain $\Omega$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet problem is solvable if $\Omega$ satisfies certain convexity condition at infinity and if $\partial \Omega$ is mean convex. We also prove that mean convexity of $\partial \Omega$ is a necessary condition, extending to unbounded domains some results that are valid on bounded ones.