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''holomorphic isomorphism, holomorphism, pseudo-conformal mapping''
 
''holomorphic isomorphism, holomorphism, pseudo-conformal mapping''
  
A generalization of the concept of a univalent [[Conformal mapping|conformal mapping]] to the case of several complex variables. A [[Holomorphic mapping|holomorphic mapping]] of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162201.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162202.png" /> is said to be a biholomorphic mapping if it is one-to-one. A biholomorphic mapping is non-degenerate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162203.png" />; its inverse mapping is also a biholomorphic mapping.
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A generalization of the concept of a univalent [[Conformal mapping|conformal mapping]] to the case of several complex variables. A [[Holomorphic mapping|holomorphic mapping]] of a domain $D\subset\mathbf C^n$ onto a domain $D'\subset\mathbf C^n$ is said to be a biholomorphic mapping if it is one-to-one. A biholomorphic mapping is non-degenerate in $D$; its inverse mapping is also a biholomorphic mapping.
  
A [[Domain of holomorphy|domain of holomorphy]] is mapped into a domain of holomorphy under a biholomorphic mapping; holomorphic, pluriharmonic and plurisubharmonic functions are also invariant under a biholomorphic mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162204.png" />, biholomorphic mappings are not conformal (except for a number of linear mappings) and the [[Riemann theorem|Riemann theorem]] is invalid for biholomorphic mappings (e.g. a ball and a polydisc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162205.png" /> cannot be biholomorphically mapped onto each other). A biholomorphic mapping of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162206.png" /> onto itself is said to be a (holomorphic) automorphism; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162207.png" />, there exist simply-connected domains without automorphisms other than the identity mapping.
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A [[Domain of holomorphy|domain of holomorphy]] is mapped into a domain of holomorphy under a biholomorphic mapping; holomorphic, pluriharmonic and plurisubharmonic functions are also invariant under a biholomorphic mapping. If $n>1$, biholomorphic mappings are not conformal (except for a number of linear mappings) and the [[Riemann theorem|Riemann theorem]] is invalid for biholomorphic mappings (e.g. a ball and a polydisc in $\mathbf C^2$ cannot be biholomorphically mapped onto each other). A biholomorphic mapping of a domain $D$ onto itself is said to be a (holomorphic) automorphism; if $n>1$, there exist simply-connected domains without automorphisms other than the identity mapping.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Concerning boundary behaviour of biholomorphic mappings the following results have been obtained. C. Fefferman's theorem: A biholomorphic mapping between strongly pseudo-convex domains with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162208.png" />-smooth boundary extends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b0162209.png" />-smoothly to a diffeomorphism between the closures of the domains, see [[#References|[a3]]]. The same result holds if the domains are only pseudo-convex and one of them satisfies condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622011.png" /> for the Bergman projection, see [[#References|[a2]]]. For strongly pseudo-convex domains with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622012.png" />-boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622014.png" /> extendability was obtained (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622016.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622017.png" /> otherwise) by L. Lempert and by S. Pinčuk. For (weakly) pseudo-convex domains with real-analytic boundary one has even holomorphic extension to a neighbourhood of the closure, see [[#References|[a1]]]. Similar results were obtained for proper holomorphic mappings.
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Concerning boundary behaviour of biholomorphic mappings the following results have been obtained. C. Fefferman's theorem: A biholomorphic mapping between strongly pseudo-convex domains with $C^\infty$-smooth boundary extends $C^\infty$-smoothly to a diffeomorphism between the closures of the domains, see [[#References|[a3]]]. The same result holds if the domains are only pseudo-convex and one of them satisfies condition $R$ for the Bergman projection, see [[#References|[a2]]]. For strongly pseudo-convex domains with $C^k$-boundary, $k>2$, $C^{k-1-\epsilon}$ extendability was obtained ($\epsilon>0$ if $k=2,3,\dots,$ $\epsilon=0$ otherwise) by L. Lempert and by S. Pinčuk. For (weakly) pseudo-convex domains with real-analytic boundary one has even holomorphic extension to a neighbourhood of the closure, see [[#References|[a1]]]. Similar results were obtained for proper holomorphic mappings.
  
A biholomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622018.png" /> is proper (i.e. the pre-image of a compact set is compact), since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622019.png" /> is continuous. Riemann's theorem does not hold in the following sense: There is no proper holomorphic mapping from the polydisc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622020.png" /> onto the ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622021.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622022.png" />, cf. [[#References|[a4]]]. Thus, function theory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622024.png" />, is strongly related to the domain of definition of the functions. For function theory in the (unit) ball of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622025.png" /> see [[#References|[a5]]]; for function theory in polydiscs see [[#References|[a6]]]. For entire holomorphic mappings and their value distribution see [[#References|[a7]]].
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A biholomorphic mapping $f$ is proper (i.e. the pre-image of a compact set is compact), since $f^{-1}$ is continuous. Riemann's theorem does not hold in the following sense: There is no proper holomorphic mapping from the polydisc in $\mathbf C^n$ onto the ball in $\mathbf C^m$ for any $n,m>1$, cf. [[#References|[a4]]]. Thus, function theory in $\mathbf C^n$, $n\geq1$, is strongly related to the domain of definition of the functions. For function theory in the (unit) ball of $\mathbf C^n$ see [[#References|[a5]]]; for function theory in polydiscs see [[#References|[a6]]]. For entire holomorphic mappings and their value distribution see [[#References|[a7]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Baouendi,  H. Jacobowitz,  F. Trèves,  "On the analyticity of CR mappings"  ''Ann. of Math.'' , '''122'''  (1985)  pp. 365–400</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  St. Bell,  "Biholomorphic mappings and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622026.png" /> problem"  ''Ann. of Math.'' , '''114'''  (1981)  pp. 103–113</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Fefferman,  "The Bergman kernel and biholomorphic mappings of pseudoconvex domains"  ''Inv. Math.'' , '''26'''  (1974)  pp. 1–65</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.G. Krantz,  "Function theory of several complex variables" , Wiley  (1982)  pp. Chapt. 10</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016220/b01622027.png" />" , Springer  (1980)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Ph.A. Griffiths,  "Entire holomorphic mappings in one and several variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Baouendi,  H. Jacobowitz,  F. Trèves,  "On the analyticity of CR mappings"  ''Ann. of Math.'' , '''122'''  (1985)  pp. 365–400</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  St. Bell,  "Biholomorphic mappings and the $\partial$ problem"  ''Ann. of Math.'' , '''114'''  (1981)  pp. 103–113</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Fefferman,  "The Bergman kernel and biholomorphic mappings of pseudoconvex domains"  ''Inv. Math.'' , '''26'''  (1974)  pp. 1–65</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.G. Krantz,  "Function theory of several complex variables" , Wiley  (1982)  pp. Chapt. 10</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in $\mathbf C^n$" , Springer  (1980)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Ph.A. Griffiths,  "Entire holomorphic mappings in one and several variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR></table>

Latest revision as of 07:43, 23 August 2014

holomorphic isomorphism, holomorphism, pseudo-conformal mapping

A generalization of the concept of a univalent conformal mapping to the case of several complex variables. A holomorphic mapping of a domain $D\subset\mathbf C^n$ onto a domain $D'\subset\mathbf C^n$ is said to be a biholomorphic mapping if it is one-to-one. A biholomorphic mapping is non-degenerate in $D$; its inverse mapping is also a biholomorphic mapping.

A domain of holomorphy is mapped into a domain of holomorphy under a biholomorphic mapping; holomorphic, pluriharmonic and plurisubharmonic functions are also invariant under a biholomorphic mapping. If $n>1$, biholomorphic mappings are not conformal (except for a number of linear mappings) and the Riemann theorem is invalid for biholomorphic mappings (e.g. a ball and a polydisc in $\mathbf C^2$ cannot be biholomorphically mapped onto each other). A biholomorphic mapping of a domain $D$ onto itself is said to be a (holomorphic) automorphism; if $n>1$, there exist simply-connected domains without automorphisms other than the identity mapping.

References

[1] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)


Comments

Concerning boundary behaviour of biholomorphic mappings the following results have been obtained. C. Fefferman's theorem: A biholomorphic mapping between strongly pseudo-convex domains with $C^\infty$-smooth boundary extends $C^\infty$-smoothly to a diffeomorphism between the closures of the domains, see [a3]. The same result holds if the domains are only pseudo-convex and one of them satisfies condition $R$ for the Bergman projection, see [a2]. For strongly pseudo-convex domains with $C^k$-boundary, $k>2$, $C^{k-1-\epsilon}$ extendability was obtained ($\epsilon>0$ if $k=2,3,\dots,$ $\epsilon=0$ otherwise) by L. Lempert and by S. Pinčuk. For (weakly) pseudo-convex domains with real-analytic boundary one has even holomorphic extension to a neighbourhood of the closure, see [a1]. Similar results were obtained for proper holomorphic mappings.

A biholomorphic mapping $f$ is proper (i.e. the pre-image of a compact set is compact), since $f^{-1}$ is continuous. Riemann's theorem does not hold in the following sense: There is no proper holomorphic mapping from the polydisc in $\mathbf C^n$ onto the ball in $\mathbf C^m$ for any $n,m>1$, cf. [a4]. Thus, function theory in $\mathbf C^n$, $n\geq1$, is strongly related to the domain of definition of the functions. For function theory in the (unit) ball of $\mathbf C^n$ see [a5]; for function theory in polydiscs see [a6]. For entire holomorphic mappings and their value distribution see [a7].

References

[a1] M.S. Baouendi, H. Jacobowitz, F. Trèves, "On the analyticity of CR mappings" Ann. of Math. , 122 (1985) pp. 365–400
[a2] St. Bell, "Biholomorphic mappings and the $\partial$ problem" Ann. of Math. , 114 (1981) pp. 103–113
[a3] C. Fefferman, "The Bergman kernel and biholomorphic mappings of pseudoconvex domains" Inv. Math. , 26 (1974) pp. 1–65
[a4] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 10
[a5] W. Rudin, "Function theory in the unit ball in $\mathbf C^n$" , Springer (1980)
[a6] W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
[a7] Ph.A. Griffiths, "Entire holomorphic mappings in one and several variables" , Annals Math. Studies , 85 , Princeton Univ. Press (1976)
How to Cite This Entry:
Biholomorphic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Biholomorphic_mapping&oldid=14872
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article