On the geometry of the Heisenberg group with a balanced metric

Abstract

We study the geometry of the Heisenberg group ${\mathrm{Nil}}_3$ with a balanced metric, the sum of the left and right invariant metrics. We prove that with this metric, ${\mathrm{Nil}}_3$ splits as a Riemannian product $\mathbb{T}\times \mathbb{Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of ${\mathrm{Nil}}_3$. So we prove the existence of complete properly embedded minimal surfaces in ${\mathrm{Nil}}_3$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. We also show the existence of complete properly embedded minimal surfaces foliating an open set of ${\mathrm{Nil}}_3$ having as boundary a given curve Γ 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 $\mathbb{T}\setminus \mathrm{\Gamma }$.