# Hartogs extension theorem

###### Theorem.

Suppose $V$ is an analytic variety in an open set $U\mathrm{\subset}{\mathrm{C}}^{n}$
($n\mathrm{\ge}\mathrm{2}$)
of dimension^{} at most $n\mathrm{-}\mathrm{2}$ and suppose that $f\mathrm{:}U\mathrm{\backslash}V\mathrm{\to}\mathrm{C}$ is holomorphic. Then there exists a unique holomorphic
extention of $f$ to all of $U$.

Note that when $V$ is 0 dimensional (a point) then this is just a special
case of the Hartogs’ phenomenon. Also note the similarity to Riemann’s
removable singularity^{} theorem in several variables, where however we also assume that $f$ is locally bounded.

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
- 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | Hartogs extension theorem |
---|---|

Canonical name | HartogsExtensionTheorem |

Date of creation | 2013-03-22 15:34:54 |

Last modified on | 2013-03-22 15:34:54 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 32H02 |