Biconservative surfaces with constant mean curvature in Lorentzian space forms

In this paper, we consider biconservative and biharmonic isometric immersions into the 4-dimensional Lorentzian space form \({\mathbb {L}}^4(\delta )\) ${\mathbb{L}}^4(\delta )$ with constant sectional curvature \(\delta \) $\delta$ . We obtain some local classifications of biconservative CMC surfaces in \({\mathbb {L}}^4(\delta )\) ${\mathbb{L}}^4(\delta )$ . Further, we get complete classification of biharmonic CMC surfaces in the de Sitter 4-space. We also proved that there is no biharmonic CMC surface in the anti-de Sitter 4-space. Further, we get the classification of biconservative, quasi-minimal surfaces in Minkowski-4 space.