# Global properties of biconservative surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$

## TL;DR

This paper surveys recent findings on biconservative surfaces in 3D space forms, focusing on their local and global properties, and classifies all complete non-constant mean curvature biconservative surfaces in Euclidean and spherical spaces.

## Contribution

It provides a comprehensive survey and classification of non-CMC biconservative surfaces in $
^3$ and $S^3$, highlighting new global and local geometric properties.

## Key findings

- All non-CMC complete biconservative surfaces in $
^3$ and $S^3$ are characterized.
- The paper emphasizes differences between the cases $c=0$ and $c=1$.
- It discusses both intrinsic and extrinsic properties of these surfaces.

## Abstract

We survey some recent results on biconservative surfaces in $3$-dimensional space forms $N^3(c)$ with a special emphasis on the $c=0$ and $c=1$ cases. We study the local and global properties of such surfaces, from extrinsic and intrinsic point of view. We obtain all non-$CMC$ complete biconservative surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$.