## Generalized Calabi's correspondence and complete spacelike surfaces    [PDF]

Hojoo Lee, José M. Manzano
Extending Calabi's correspondence between minimal graphs in the Euclidean space $\mathbb{R}^3$ and maximal graphs in the Lorentz-Minkowski spacetime $\mathbb{L}^3$ to a wide class of 3-manifolds carrying a unit Killing vector field, we construct a twin correspondence between graphs with prescribed mean curvature $H$ in the Riemannian Generalized Bianchi-Cartan-Vranceanu (GBCV) space $\mathbb{E}^3(M,\tau)$ and spacelike graphs with prescribed mean curvature $\tau$ in the GBCV spacetime $\mathbb{L}^{3}(M, H)$. For instance, the prescribed mean curvature equation in $\mathbb{L}^3$ can be transformed into the minimal surface equation in the generalized Heisenberg space with prescribed bundle curvature. We present several applications of the twin correspondence and study the moduli space of complete spacelike surfaces in the GBCV spacetimes.
View original: http://arxiv.org/abs/1301.7241