Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic and exterior Dirichlet problems for the minimal surface
  equation in the Heisenberg group with a balanced metric   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 $\mathbb{T}\backslash\Gamma$.

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The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: eess