On the existence of closed biconservative surfaces in space forms

Biconservative surfaces of Riemannian $3$-space forms $N^3(\rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3 \kappa_1 + \kappa_2 = 0$ between their principal curvatures $\kappa_1$ and $\kappa_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round $3$-sphere, $\mathbb{S}^3(\rho)$. However, none of these closed surfaces is embedded in $\mathbb{S}^ (\rho)$.