BCV spaces are a family of 3-dimensional Riemannian manifolds which include six of Thurston’s eight geometries. In this paper, we give a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional BCV space by proving that such biharmonic maps exist only in the cases of \(H^2\times \mathbb {R}\rightarrow \mathbb {R}^2\) , or \({\widetilde{SL}}(2,\mathbb {R})\rightarrow \mathbb {R}^2\) . In each of these two cases, we are able to construct a family of infinitely many proper biharmonic Riemannian submersions. Our results, on one hand, extend a previous result of the authors which gave a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional space form, and, on the other hand, they can be viewed as the dual study of biharmonic surfaces (i.e., biharmonic isometric immersions) in a BCV space studied in some recent literature.
