  We study existence and structure of $P-$area minimizing surfaces in the
Heisenberg group under Dirichlet and Neumann boundary conditions. We show that
there exists an underlying vector field $N$ that characterized existence and
structure of $P$-area minimizing surfaces. This vector field exists even if
there is no $P$-area minimizing surface satisfying the prescribed boundary
conditions. We prove that if $\partial \Omega$ satisfies a so called Barrier
condition, it is sufficient to guarantee existence of such surfaces. Our
approach is completely different from previous methods in the literature and
makes major progress in understanding existence of $P$-area minimizing
surfaces.
