## Abstract

In the paper [13] we proved that the only stable C^{2} minimal surfaces in the first Heisenberg group ℍ^{1} which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in [13] to C^{2} complete embedded minimal surfaces in ℍ^{1} with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane. This result represents a sub-Riemannian counterpart of the classical theorems of Fischer-Colbrie and Schoen, [16], and do Carmo and Peng, [15], and answers a question posed by Lei Ni.

Original language | English (US) |
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Pages (from-to) | 563-594 |

Number of pages | 32 |

Journal | Indiana University Mathematics Journal |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

## Keywords

- Bernstein problem
- Heisenberg group
- Intrinsic graph
- Minimal surfaces

## ASJC Scopus subject areas

- Mathematics(all)

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