On the uniqueness of complete biconservative surfaces in $3$-dimensional space forms

TL;DR

This paper proves the uniqueness of complete, simply connected, non-constant mean curvature biconservative surfaces in 3D space forms, addressing a fundamental question about their classification.

Contribution

It establishes the uniqueness of such surfaces, providing a definitive answer to the previously open question in differential geometry.

Findings

Uniqueness of complete non-CMC biconservative surfaces in 3D space forms.

Resolution of the open problem regarding their classification.

Supports the understanding of stress-bienergy tensor divergence-free surfaces.

Abstract

Biconservative surfaces are surfaces with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative surfaces in $3$-dimensional space forms were constructed working in extrinsic and intrinsic ways. Then, one raises the question of the uniqueness of such surfaces. In this paper we give a positive answer to this question.