Generalized Calabi correspondence and complete spacelike surfaces

Abstract

Extending Calabi's correspondence between minimal graphs in the Euclidean spaceR 3 and maximal graphs in the Lorentz-Minkowski spacetimeL 3 to a wide class of 3-manifolds carrying a unit Killing vec- tor field, we construct a twin correspondence between graphs with pre- scribedmeancurvatureHintheRiemannianGeneralizedBianchi-Cartan- Vranceanu (GBCV) spaceE 3 (M,�) and spacelike graphs with prescribed mean curvaturein the GBCV spacetimeL 3 (M,H). For instance, the prescribed mean curvature equation inL 3 can be transformed into the minimal surface equation in the generalized Heisenberg space with pre- scribed bundle curvature. We present several applications of the twin correspondence and study the moduli space of complete spacelike sur- faces in the GBCV spacetimes. 1. Motivationandmainresults