The Codazzi equation for surfaces

In this paper we study the classical Codazzi equation in space forms from an abstract point of view, and use it as an analytic tool to derive global results for surfaces in different ambient spaces. In particular, we study the existence of holomorphic quadratic differentials, the uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces.