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

TL;DR

This paper proves that the only complete biconservative surfaces in three-dimensional Euclidean space are either constant mean curvature surfaces or specific surfaces of revolution, with compact ones being round spheres.

Contribution

It establishes the uniqueness of complete biconservative surfaces in ^3, identifying them as either CMC surfaces or particular revolution surfaces, and classifies compact cases as spheres.

Findings

Complete biconservative surfaces are either CMC or certain revolution surfaces.

Compact biconservative surfaces are round spheres.

No other complete biconservative surfaces exist in ^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.