Codazzi surfaces in 4-manifolds

We study the geometry of Codazzi surfaces immersed in 4-manifolds with mean curvature vector satisfying a differential inequality that generalizes the condition of having parallel mean curvature. In this way, we extend some rigidity results obtained in the past by several authors. By similar techniques, we also study the geometry of smooth maps between Riemann surfaces whose tension field is suitably controlled by the energy density.