View source for Biholomorphic 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.

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>



====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.

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]]].

====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>