## Abstract

In the first Heisenberg group ℍ^{1} with its sub-Riemannian struc-ture generated by the horizontal subbundle, we single out a class of C^{2} non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deforma-tions for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C^{2} entire H-minimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in ℍ^{1} the only stable C2 H- minimal entire graphs, with empty characteristic locus, are the vertical planes.

Original language | English (US) |
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Pages (from-to) | 251-295 |

Number of pages | 45 |

Journal | Journal of Differential Geometry |

Volume | 81 |

Issue number | 2 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology