Surfaces in S{double-struck}~2×R{double-struck} and H{double-struck}~2×R{double-struck} with holomorphic Abresch-Rosenberg differential

摘要

We describe all surfaces in S{double-struck}~2×R{double-struck} and H{double-struck}~2×R{double-struck} 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 H{double-struck}2. The non-rotational surfaces in H{double-struck}~2×R{double-struck} extend to regular graphs over H{double-struck}2; these are new examples of complete surfaces in H{double-struck}~2×R{double-struck} with constant Gaussian curvature Kε(-1,0).