Complete biconservative surfaces in ${\mathbb{R}}^3$ and ${\mathbb{S}}^3$

Abstract

In this paper we consider the complete biconservative surfaces in Euclidean space ${\mathbb{R}}^3$ and in the unit Euclidean sphere ${\mathbb{S}}^3$. 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 ${\mathbb{R}}^3$ and ${\mathbb{S}}^3$. Moreover, such immersions are explicitly described.

29 pages, 6 figures

References in corpus (2)

• Properties of biharmonic submanifolds in spheres
• Complete classification of biconservative hypersurfaces with diagonalizable shape operator in Minkowski 4-space

Cited by in corpus (2)

• On the Existence of Closed Biconservative Surfaces in Space Forms
• Complete biconservative surfaces in the hyperbolic space ${\mathbb{H}}^3$