Marginally trapped ruled surfaces and their Gauss map in Minkowski space (Q6619698)

An isometric immersion \(x\) of a Riemannian manifold \(M\) into a Euclidean space \(\mathbb{E}^{m}\) is said to be biharmonic if it satisfies \(\Delta^{2}x = 0\), where \(\Delta\) denotes the Laplace operator defined on \(M\). Chen conjectured that biharmonic submanifolds of Euclidean spaces are minimal and proved it for submanifolds of \(\mathbb{E}^{3}\). The same notion can be extended to pseudo-Euclidean spaces \(\mathbb{E}^{m}_{s}\): in this setting \textit{B.-Y. Chen} et al. [Bull. Aust. Math. Soc. 42, No. 3, 447--453 (1990; Zbl 0704.53003)] showed that biharmonic surfaces in pseudo-Euclidean 3-spaces are minimal and that there exist proper biharmonic surfaces in 4-dimensional pseudo-Euclidean spaces \(\mathbb{E}^{4}_{s}\) with \(s\in \{1,2,3\}\). This means that Chen's conjecture does not extend to submanifolds in pseudo-Euclidean space.\N\NIn general relativity, a marginally trapped surface in pseudo-Euclidean space is a Riemannian surface whose mean curvature vector field is null at every point of the surface.\N\NThis paper investigates biharmonic marginally trapped ruled surfaces in \(\mathbb{E}^{m}_{1}\) and finds new examples disproving Chen's conjecture in this setting.