On the Existence of Closed Biconservative Surfaces in Space Forms

Abstract

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, $S^3(\rho )$. However, none of these closed surfaces is embedded in $S^3(\rho )$.

To appear in Communications in Analysis and Geometry

Cited by in corpus (2)

• Closed Biconservative Hypersurfaces in Spheres
• Biconservative hypersurfaces with constant scalar curvature in space forms