Communications in Analysis and Geometry

Volume 31 (2023)

Number 2

On the existence of closed biconservative surfaces in space forms

Pages: 291 – 319



S. Montaldo (Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Italy)

A. Pámpano (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Tx., U.S.A.)


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 $\mathbb{S}^ (\rho)$.

The authors’ work was partially supported by Fondazione di Sardegna and Regione Autonoma della Sardegna (Project GESTA). The second author has been partially supported by MINECO-FEDER grant PGC2018-098409-B-100, Gobierno Vasco grant IT1094-16 and Programa Predoctoral de Formación de Personal Investigador No Doctor del Gobierno Vasco, 2015.

Received 11 January 2019

Accepted 7 September 2020

Published 6 December 2023