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

Abstract

Given an unbounded domain Ω 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 ≤ −1 this Dirichlet problem is solvable if Ω satisfies certain convexity condition at infinity and if ∂Ω is mean convex. We also prove that mean convexity of ∂Ω is a necessary condition, extending to unbounded domains some results that are valid on bounded ones.

Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

“…14.10 p. 347]). Afterward M. Telichevesky [19,Lemma. 11 p. 250] extended the result for the minimal vertical equation in M × R. We will use some of the ideas of these works.…”

“…Finally, M. Telichevesky [19,Th. 6 p. 246] proved that if M is a Hadamard manifold whose sectional curvature is bounded above by −1, then mean convexity is a necessary condition for the existence of a vertical minimal graphs in To the best of our knowledge, no other non-existence result and Serrin-type solvability criterion have been proved in settings different from the Euclidean one.…”

“…Finally, M. Telichevesky [19,Th. 6 p. 246] proved that if M is a Hadamard manifold whose sectional curvature is bounded above by −1, then mean convexity is a necessary condition for the existence of a vertical minimal graphs in M × R over a domain Ω of M possibly unbounded.…”

Survey on the asymptotic Dirichlet problem for the minimal surface equation

“…Telichevesky [48] considered the Dirichlet problem on unbounded domains Ω proving the existence of solutions provided that K M ≤ −1, the ordinary boundary of Ω is mean convex and that Ω satisfies the SC condition at infinity. The SC condition was studied by Casteras, Holopainen and Ripoll also in [13] and they proved that the manifold M satisfies the SC condition under very general curvature assumptions.…”

PDE and hypersurfaces with prescribed mean curvature

“…This result generalizes the existence part in Theorem B stated in the introduction. In the case where M is a Hadamard manifold whose sectional curvature is bounded above by −1, then the mean convexity condition is sharp due to a work of M. Telichevesky [26,Th. 6 p. 246].…”