On the uniqueness of complete biconservative surfaces in ${\mathbb{R}}^3$

Abstract

We study the uniqueness of complete biconservative surfaces in the Euclidean space ${\mathbb{R}}^3$, and prove that the only complete biconservative regular surfaces in ${\mathbb{R}}^3$ are either $CMC$ or certain surfaces of revolution. In particular, any compact biconservative regular surface in ${\mathbb{R}}^3$ is a round sphere.

15 pages, 2 figures