## Abstract

It follows from the classical argument principle that if a holomorphic mapping f from the unit disc δ to C or, more generally, to Cn, smooth in the closed disc, is homologically trivial on the unit circle (i.e. _{H1}(γ) = 0, γ = f(^{S1}), which is equivalent to either γ being a point or ∂γ ≠ θ ), then f = const, i.e. the image of the unit disc degenerates to a point. We establish a parametric version of this fact, for a variety of holomorphic mappings from δ to Cn in place of a single mapping. We find conditions for a holomorphic mapping of the unit disc, depending on additional real parameters, under which homological triviality of the boundary image implies collapse of the dimension of the image of the interior. As an application, we obtain estimates of dimensions of complex tangent bundles of real submanifolds in Cn, in terms of zero moment conditions on families of closed curves covering the manifold. Applying this result to the graphs of functions, we obtain solution of several known problems about characterization of holomorphic CR functions in terms of moment conditions on families of curves.

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

Pages (from-to) | 38-85 |

Number of pages | 48 |

Journal | Advances in Mathematics |

Volume | 255 |

DOIs | |

State | Published - 1 Apr 2014 |

### Bibliographical note

Funding Information:This work was partially supported by ISF (Israel Science Foundation) , Grant 688/08 . Some of this research was done as a part of European Networking Program HCAA.

### Funding

This work was partially supported by ISF (Israel Science Foundation) , Grant 688/08 . Some of this research was done as a part of European Networking Program HCAA.

Funders | Funder number |
---|---|

Israel Science Foundation | 688/08 |

## Keywords

- Analytic disc
- Argument principle
- CR function
- CR manifold
- Holomorphic function
- Homology