Maximal surfaces in Lorentzian Heisenberg space

Abstract

We build the twin correspondence between surfaces of constant mean curvature in R 3 and maximal surfaces in the Lorentzian Heisenberg space Nil 1 3 ( τ ) . We prove that Gauss maps of the maximal surfaces in Nil 1 3 ( τ ) are harmonic maps into S 2 and that some perturbed Hopf differentials on the maximal surfaces in Nil 1 3 ( τ ) are holomorphic. We solve the Calabi–Bernstein problem for maximal surfaces in Nil 1 3 ( τ ) .