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 ${\mathrm{Nil}}_3$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T}\times \mathbb{Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\mathrm{Nil}$ It is then proved the existence of complete properly embedded minimal surfaces in ${\mathrm{Nil}}_3$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. It is also proved the existence of complete properly embedded minimal surfaces foliating an open set of ${\mathrm{Nil}}_3$ having as boundary a given curve $\Gamma$ in $\mathbb{T},$ satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of $\mathbb{T}\backslash \Gamma$.