### Abstract

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 R^{3}
_{1}is called of mixed type if it changes causal type from space-like to time-like. In R^{3}
_{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.

Original language | English |
---|---|

Pages (from-to) | 801-837 |

Number of pages | 37 |

Journal | Quarterly Journal of Mathematics |

Volume | 67 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2016 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Quarterly Journal of Mathematics*,

*67*(4), 801-837. https://doi.org/10.1093/qmath/haw038