Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three

Source abstract

We study the total lifts of curves by means of a submersion π : M s 3 → B r 2 \pi :M_s^3\rightarrow B_r^2 that satisfy the condition Δ H = λ H \Delta H=\lambda H analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case λ = 0 \lambda =0 (biharmonic lifts). Moreover, we obtain a biharmonic lift in R 3 {\mathbb R}^3 by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for R 3 {\mathbb R}^3 with a non-flat Riemannian metric.