Biharmonic submanifolds in nonflat Lorentz 3-space forms (Q2891970)

A map \(\varphi: M^n \to N^n\) is called biharmonic if it is a critical point of the bienergy given by the tension field of \(\varphi\) under compactly supported variations. Harmonic maps are automatically biharmonic and nonharmonic biharmonic maps are called proper biharmonic maps. Proper biharmonic maps in the pseudo-Euclidean \(3\)-space have been classified. In particular, it is known that there exists no proper biharmonic surface is pseudo-Euclidean \(3\)-space.NEWLINENEWLINEIn this paper, the author classifies proper biharmonic curves and surfaces in de Sitter \(3\)-space and anti-de Sitter \(3\)-space. He shows that there are five types of proper biharmonic curves in de Sitter \(3\)-space and anti-de Sitter \(3\)-space, respectively. The author also proves that if \(M^2\) is a pseudo-Riemannian surface in a Lorentz \(3\)-space form of constant sectional curvature \(1\) or \(-1\) which is proper biharmonic, then \(M\) is congruent to either \(S^2_1(1) \subset S^3_1(1)\), \(H^2_1(-1) \subset H^3_1(-1)\), or a \(B\)-scroll over a null curve of constant Gaussian curvature \(2\) in \(S^3_1(1)\), where \(S^n_1(1)\) and \(H^n_1(-1)\) are de Sitter space and anti-de Sitter space, respectively. In particular, contrast to the case of pseudo-Euclidean \(3\)-space, there exist proper biharmonic pseudo-Riemannian surfaces in nonflat Lorentz \(3\)-space forms.