Surfaces in S2×R and H2×R with holomorphic Abresch-Rosenberg differential

We describe all surfaces in S2×R and H2×R with holomorphic Abresch-Rosenberg differential (originally defined in Abresch and Rosenberg, 2004 [1]) and non-constant mean curvature. We prove that the horizontal slices of these surfaces are the level curves of the mean curvature H, whose projections determine either a polar system of geodesic rays and circles in the base (rotational surfaces) or an orthogonal system of ultra-parallel geodesics and equidistant curves in H2. The non-rotational surfaces in H2×R extend to regular graphs over H2; these are new examples of complete surfaces in H2×R with constant Gaussian curvature K∈(−1,0).