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

TL;DR

This paper classifies and explicitly describes complete biconservative surfaces in Euclidean space and the sphere, focusing on their geometric properties and the conditions for their existence.

Contribution

It characterizes complete biconservative surfaces in $ ^3$ and $S^3$, providing explicit descriptions and identifying the simply connected ones that admit such immersions.

Findings

Complete classification of biconservative surfaces in $ ^3$ and $S^3$

Explicit descriptions of these surfaces

Identification of simply connected, complete examples

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.