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Difference between revisions of "Behnke-Stein theorem"

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The union of domains of holomorphy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154302.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154303.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154304.png" />, is also a domain of holomorphy. The Behnke–Stein theorem is valid not only in the complex Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154305.png" />, but also on any [[Stein manifold|Stein manifold]]. If the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154306.png" /> is not monotone increasing by imbedding, the theorem is not valid; e.g. the union of the two domains of holomorphy
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The union of [[domain of holomorphy |domains of holomorphy]] $G_k \subset \mathbf{C}^n$, $k=1,2,\ldots$ for which $G_k \subseteq G_{k+1}$ for all $k$, is also a domain of holomorphy. The Behnke–Stein theorem is valid not only in the complex Euclidean space $\mathbf{C}^n$, but also on any [[Stein manifold]]. If the sequence $G_k$ is not monotone increasing under imbedding, the theorem is not valid; e.g. the union of the two domains of holomorphy
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154307.png" /></td> </tr></table>
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G_1 = \{ (z_1,z_2) : |z_1| <1 \,,\ |z_2| < 2 \}
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and
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G_2 = \{ (z_1,z_2) : |z_1| <2 \,,\ |z_2| < 1 \}
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in $\mathbf{C}^2$ is not a domain of holomorphy.
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154308.png" /></td> </tr></table>
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Behnke,  K. Stein, "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität" ''Math. Ann.'' , '''116'''  (1938)  pp. 204–216 {{DOI|10.1007/BF01597355}} {{ZBL|0020.37803}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> Hans Grauert, Reinhold Remmert, "Theory of Stein spaces" (Tr. Alan Huckleberry) Grundlehren der mathematischen Wissenschaften '''236''' Springer (1979, repr.2008) {{ISBN|3-540-00373-8}} {{ZBL|0433.32007}}
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</table>
  
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in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015430/b0154309.png" /> is not a domain of holomorphy.
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{{TEX|done}}
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Behnke,  K. Stein,  "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität"  ''Math. Ann.'' , '''116'''  (1938)  pp. 204–216</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 

Latest revision as of 16:56, 25 November 2023

The union of domains of holomorphy $G_k \subset \mathbf{C}^n$, $k=1,2,\ldots$ for which $G_k \subseteq G_{k+1}$ for all $k$, is also a domain of holomorphy. The Behnke–Stein theorem is valid not only in the complex Euclidean space $\mathbf{C}^n$, but also on any Stein manifold. If the sequence $G_k$ is not monotone increasing under imbedding, the theorem is not valid; e.g. the union of the two domains of holomorphy $$ G_1 = \{ (z_1,z_2) : |z_1| <1 \,,\ |z_2| < 2 \} $$ and $$ G_2 = \{ (z_1,z_2) : |z_1| <2 \,,\ |z_2| < 1 \} $$ in $\mathbf{C}^2$ is not a domain of holomorphy.

References

[1] H. Behnke, K. Stein, "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität" Math. Ann. , 116 (1938) pp. 204–216 DOI 10.1007/BF01597355 Zbl 0020.37803
[2] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[a1] Hans Grauert, Reinhold Remmert, "Theory of Stein spaces" (Tr. Alan Huckleberry) Grundlehren der mathematischen Wissenschaften 236 Springer (1979, repr.2008) ISBN 3-540-00373-8 Zbl 0433.32007
How to Cite This Entry:
Behnke-Stein theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Behnke-Stein_theorem&oldid=22077
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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