The purpose of this paper is to classify nonharmonic biharmonic curves and surfaces in de Sitter 3-space and anti-de Sitter 3-space.2010 Mathematics subject classification: primary 53C42; secondary 53B25.

A natural generalization of the class of minimal submanifolds from the variational point of view is the one of biharmonic submanifolds (see, [9]), which was introduced by Eells and Sampson [9]. Thus, it is worthwhile and interesting to investigate biharmonic Lagrangian submanifolds. For recent developments in the study of biharmonic submanifolds see, for example, [2], [3], [10] and [15]. In this paper, biharmonic Lagrangian surfaces of constant mean curvature in complex space forms are classified. In particular, we obtain new examples of marginally trapped Lagrangian surfaces in an indefinite complex Euclidean plane. This implies that the classification of marginally trapped biharmonic surfaces due to Chen and Ishikawa [6] is not complete.

A biminimal submanifold is a critical point of the bienergy functional for any normal variation. We give a complete description of all nonminimal biminimal Legendrian submanifolds in Sasakian space forms which are Legendrian H‐umbilical.

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Abstract. We investigate quasi-minimal Lagrangian surfaces whose mean curvature vectors sire eigenvectors of the Laplace operator.