  We compute a Simons' type formula for the stress-energy tensor of biharmonic
maps from surfaces. Specializing to Riemannian immersions, we prove several
rigidity results for biharmonic CMC surfaces, putting in evidence the influence
of the Gaussian curvature on pseudo-umbilicity. Finally, the condition of
biharmonicity is shown to enable an extension of the classical Hopf theorem to
CMC surfaces in any ambient Riemannian manifold.
