The Codazzi Equation for Surfaces

TL;DR

This paper develops an abstract framework for the Codazzi equation on surfaces and uses it to derive new global results and generalizations for surfaces in space forms, impacting the study of Weingarten surfaces.

Contribution

It introduces an abstract theory for the Codazzi equation and applies it to generalize classical surface theorems and analyze Weingarten surfaces in space forms.

Findings

Generalized classical theorems depending on the Codazzi equation

Established existence of holomorphic quadratic differentials

Proved uniqueness and height estimates for certain surfaces

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 ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. 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.