Hartogs extension theorem

Theorem.

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

Note that when $V$ is 0 dimensional (a point) then this is just a specialcase of the Hartogs’ phenomenon. Also note the similarity to Riemann’sremovable 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.

单词	HartogsExtensionTheorem
释义	Hartogs extension theorem Theorem. Suppose $V$ is an analytic variety in an open set $U\\subset{\\mathbb{C}}^{n}$ ( $n\\geq 2$ )of dimension at most $n-2$ and suppose that $f\\colon U\\backslash V\\to{\\mathbb{C}}$ is holomorphic. Then there exists a unique holomorphicextention of $f$ to all of $U$ . Note that when $V$ is 0 dimensional (a point) then this is just a specialcase of the Hartogs’ phenomenon. Also note the similarity to Riemann’sremovable 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.
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