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abstract: 'In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms $\r^3$, $\s^3$ and $\h^3$. We give essentially sharp generalizations of some classical theorems of surface theory that mainly depend on the Codazzi equation, and we apply them to the study of Weingarten surfaces in space forms. In particular, we study existence of holomorphic quadratic differentials, uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces.'
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[**The Codazzi Equation for Surfaces**]{}\

[**Juan A. Aledo$^a$, José M. Espinar${}^{b}$ and José A. Gálvez${}^{c}$**]{}

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[${}^a$Departamento de Matemáticas, Universidad de Castilla-La Mancha, EPSA, 02071 Albacete, Spain; e-mail: JuanAngel.Aledo@uclm.es\
${}^b$Institut de Mathématiques, Université Paris VII, 175 Rue du Chevaleret, 75013 Paris, France; e-mail: jespinar@ugr.es\
${}^c$Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain; e-mail: jagalvez@ugr.es]{}

Introduction
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The *Codazzi equation* for an immersed surface ${\Sigma}$ in $\r^3$ yields $$\label{ecod}
\nabla_XSY-\nabla_YSX-S[X,Y]=0,\qquad X,Y\in{\mathfrak{X}}({\Sigma}).$$ Here $\nabla$ is the Levi-Civita connection of the first fundamental form $I$ of ${\Sigma}$ and $S$ is the shape operator, defined by 