On the geometry of the Heisenberg group with a balanced metric

We study the geometry of the Heisenberg group Nil3 with a balanced metric, the sum of the left and right invariant metrics. We prove that with this metric, Nil3 splits as a Riemannian product T×Z, where T is a totally geodesic surface and Z the center of Nil3. So we prove the existence of complete properly embedded minimal surfaces in Nil3 by solving the asymptotic Dirichlet problem for the minimal surface equation on T. We also show the existence of complete properly embedded minimal surfaces foliating an open set of Nil3 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.