Asymptotic and exterior Dirichlet problems for the minimal surface equation in the Heisenberg group with a balanced metric

Abstract

It is proved that the Heisenberg group Nil 3 with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product T × Z, where T is a totally geodesic surface and Z the center of Nil 3 . It is then proved the existence of complete properly embedded minimal surfaces in Nil 3 by solving the asymptotic Dirichlet problem for the minimal surface equation on T. It is also proved the existence of complete properly embedded minimal surfaces foliating an open set of Nil 3 having as boundary a given curve Γ in T, satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of T\Γ.