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 $t=xy$.