Complete biconservative surfaces in R3 and S3

In this paper we consider the complete biconservative surfaces in Euclidean space R3 and in the unit Euclidean sphere S3. Biconservative surfaces in 3-dimensional space forms are characterized by the fact that the gradient of their mean curvature function is an eigenvector of the shape operator, and we are interested in studying local and global properties of such surfaces with non-constant mean curvature function. We determine the simply connected, complete Riemannian surfaces that admit biconservative immersions in R3 and S3. Moreover, such immersions are explicitly described.