View source for Preservation of domain, principle of

A property of holomorphic functions on domains in the complex plane: the set of values of every non-constant holomorphic function on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744401.png" /> is also a domain, that is, an open connected set. The basic property here is that of openness of the image, which follows from Rouché's theorem or from the principle of the argument (cf. [[Rouché theorem|Rouché theorem]]; [[Argument, principle of the|Argument, principle of the]]). The principle of preservation of domain can be considered as a generalization of the [[Maximum-modulus principle|maximum-modulus principle]] for holomorphic functions.

The principle of preservation of domain holds for holomorphic functions on an arbitrary complex manifold: the set of values of any non-constant holomorphic function on a connected [[Complex manifold|complex manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744402.png" /> is a domain in the complex plane. It also holds for holomorphic mappings from complex manifolds into Riemann surfaces. However, holomorphic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744403.png" /> into complex manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744404.png" /> of dimension exceeding 1 need not be open, in general: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744405.png" /> is non-constant, but has rank everywhere less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744406.png" />, say, then the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744407.png" /> has no interior points, in general. Openness can also be violated in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p0744408.png" /> on sets of small dimension. For example, under the mapping

<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/p/p074/p074440/p0744409.png" /></td> </tr></table>

from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p07444010.png" /> into itself, the image is the non-open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p07444011.png" />. The principle of preservation of domain holds for holomorphic mappings if the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p07444012.png" /> be non-constant is replaced by stronger requirements, one of the simplest being the zero-dimensionality of the set of points for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074440/p07444013.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Einführung in die Funktionentheorie" , '''1–3''' , Teubner  (1965–1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>



====Comments====
In the older German literature the concept occurs as the  "GebietstreueGebietstreue"  of a holomorphic mapping. In English, one usually speaks of the open mapping principle, cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.T. Whyburn,   "Topological analysis" , Princeton Univ. Press  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M. Chirka,   "Complex analytic sets" , Kluwer  (1989)  (Translated from Russian)</TD></TR></table>