On the existence of closed biconservative surfaces in space forms

Biconservative surfaces of Riemannian 3-space forms N3(ρ), are either constant mean curvature (CMC) surfaces or rotational lin- ear Weingarten surfaces verifying the relation 3κ1 + κ2 = 0 be- tween their principal curvatures κ1 and κ2. We characterise the profile curves of the non-CMC biconservative surfaces as the crit- ical 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 bicon- servative surfaces in the round 3-sphere, S3(ρ). However, none of these closed surfaces is embedded in S3(ρ).