Entire zero-mean curvature graphs of mixed type in lorentz-minkowski 3-space

It is classically known that the only entire zero-mean curvature graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space R³ ₁is called of mixed type if it changes causal type from space-like to time-like. In R³ ₁ Osamu Kobayashi found two entire zero-mean curvature graphs of mixed type that are not planes. As far as the authors know, these two examples were the only known examples of entire zero-mean curvature graphs of mixed type without singularities. In this paper, we construct several families of real analytic entire zero-mean curvature graphs of mixed type in Lorentz-Minkowski 3-space. The entire graphs mentioned above lie in one of these classes.