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

## TL;DR

This paper investigates minimal surfaces in the Heisenberg group with a balanced metric, proving the existence of such surfaces with prescribed asymptotic and boundary conditions by solving Dirichlet problems.

## Contribution

It establishes the splitting of the Heisenberg group under a balanced metric and constructs minimal surfaces satisfying asymptotic and boundary conditions via Dirichlet problem solutions.

## Key findings

- Heisenberg group with balanced metric splits as a Riemannian product.
- Existence of complete properly embedded minimal surfaces in the group.
- Solutions to Dirichlet problems yield minimal surfaces with prescribed boundary behavior.

## Abstract

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}% \nolimits_{3}.$ It is then proved the existence of complete properly embedded minimal surfaces in $\operatorname*{Nil}\nolimits_{3}$ 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 $\operatorname*{Nil}\nolimits_{3}$ 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 $\mathbb{T}\backslash\Gamma$.