The classification of complete stable area-stationary surfaces in the   Heisenberg group $\mathbb{H}^1$
  We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$. 
Ana Hurtado, Manuel Ritor\'e and C\'esar Rosales