On the Existence of Closed Biconservative Surfaces in Space Forms

TL;DR

This paper characterizes biconservative surfaces in space forms, showing that non-CMC examples exist as closed, non-embedded surfaces in the 3-sphere, expanding understanding of their geometric properties.

Contribution

It provides a characterization of non-CMC biconservative surfaces via profile curves and proves the existence of a family of closed, non-embedded such surfaces in the 3-sphere.

Findings

Existence of a discrete biparametric family of closed non-CMC biconservative surfaces in S^3.

None of these closed surfaces are embedded in S^3.

Profile curves are critical points of a specific curvature energy.

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 $S^3(\rho)$.