Some Characterizations of Linear Weingarten Surfaces in 3-Dimensional Space Forms

We consider a class of surfaces satisfying an interesting geometric equation A∇H=kH∇H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \nabla H=kH\nabla H$$\end{document} in non-flat 3-dimensional space forms N3(c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{3}(c)$$\end{document}, where A is the shape operator, H is the mean curvature and k is a constant. This kind of surfaces are called generalized biconservative surfaces (or GB surfaces for short). We prove that every GB surface in 3-dimensional space forms is linear Weingarten, and we classify all GB surfaces in 3-dimensional sphere S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^{3}$$\end{document} and hyperbolic space H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^{3}$$\end{document}, respectively. Moreover, for a curvature energy in Riemannian 3-space forms, we show that the profile curves of rotational GB surfaces can be characterized as the critical curves.