Biharmonic Hopf Cylinders

Abstract

This paper concerns curves and surfaces, into indefinite space forms, whose mean curvature vector field is in the kernel of certain elliptic differential operators. It has been inspired by the paper of M. Barros and O.J. Garay [2], where the Riemannian version of this question is solved. We first consider the Laplacian to study indefinite submanifolds with harmonic mean curvature vector field in the normal bundle. This problem is closely related to a conjecture of B.-Y. Chen [5], on Riemannian submanifolds, stated as follows: harmonicity of the mean curvature vector field implies harmonicity of the immersion. Submanifolds with harmonic mean curvature vector field were called by Chen biharmonic submanifolds. In the realm of indefinite submanifolds, counterexamples to that conjecture have been given by the two first authors (see [1]). Biharmonic submanifolds are a special class of submanifolds for which its mean curvature vector is an eigenvector of ∆, that is, ∆H = λH for some real constant λ. First we describe the family of curves whose mean curvature vector field is proper for the Laplacian. This problem has been yet solved for Euclidean curves by M. Barros and O.J. Garay [2]. We have to think of a different Laplacian if we want to characterize curves others than those of both constant curvature and torsion. Since H is a normal vector field, it seems natural to consider the Laplacian associated to the connection in the normal bundle. Then we show that the indefinite Cornu spirals are the only non standard curves in a semi-Riemannian manifold that are biharmonic in the normal bundle. As for surfaces, we deal with the semi-Riemannian Hopf cylinders we introduced in [4]. Then we show that the biharmonicity of them strongly depends on the biharmonicity of the curves to which are associated. In fact, a non standard Hopf cylinder in H1(−1) is biharmonic in the normal bundle if and only if it is associated to a Cornu spiral in Hs(−4). Then we extend the results in [2]. The second operator considered is the Jacobi operator, which was introduced by J. Simons [9] and involves the Laplacian in the normal bundle. A normal vector field is called a Jacobi field if it belongs to the kernel of the Jacobi operator. This operator appears when one studies the second variation of the area functional for compact Riemannian minimal submanifolds. It has been recently used by Barros and Garay [3] to classify Hopf cylinders into S3 with Jacobi mean curvature vector field. In [4] we have made, following [7], a qualitative description of elastic curves into indefinite space forms to be used as a tool to find Lorentzian Willmore tori in H1(−1). Now the Jacobi operator allows us to get a characterization of elastic curves, as well as a characterization of semi-Riemannian Hopf cylinders in H1(−1), in terms of elasticae in Hs(−4), s = 0, 1. We show that a curve in an indefinite real space form has Jacobi mean curvature vector field if and only if it is curvature homothetic to a free elastica. As before, this characterization leads to find Hopf cylinders into H1(−1) with Jacobi mean curvature vector field.