View source for Boundary correspondence (under conformal mapping)

''''A property of a univalent conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172801.png" /> of a finitely-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172802.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172803.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172804.png" />-plane consisting of the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172805.png" /> can be extended to a [[Homeomorphism|homeomorphism]] between certain compactifications <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b0172809.png" />, respect- ively; that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728010.png" /> induces a homeomorphism of the boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728012.png" />. For the ordinary (Euclidean) boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728016.png" /> this property does not always hold. For example, a conformal mapping of a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728017.png" /> induces a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728019.png" /> only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017280/b01728020.png" /> is homeomorphic to a circle.

There are several known compactifications of a simply-connected domain with the property of boundary correspondence under conformal mapping. Historical precedence goes to the Carathéodory extension (see [[#References|[1]]], and also [[#References|[2]]]). It is the most intuitive and is often used in the study of conformal and other mappings. The elements of the boundary thus obtained were called prime ends by C. Carathéodory (see [[Limit elements|Limit elements]]). A theory has been developed of boundary correspondence under variable conformal mappings of simply-connected domains (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.D. Myshkis,   G.D. Suvorov,   "Conformally-invariant bicompact extensions of a plane simply-connected domain"  ''Soviet Math. Dokl.'' , '''14'''  (1973–1974)  pp. 1488–1491  ''Dokl. Akad. Nauk SSSR'' , '''212'''  (1973)  pp. 822–824</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Carathéodory,   "Über die Begrenzung einfach zusammenhängender Gebiete"  ''Math. Ann.'' , '''73'''  (1913)  pp. 323–370</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Suvorov,   "Prime ends of a sequence of plane regions converging to a nucleus"  ''Amer. Math. Soc. Transl. Ser. 2'' , '''1'''  (1955)  pp. 67–93  ''Mat. Sb.'' , '''33 (75)'''  (1953)  pp. 73–100</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1965–1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.D. Suvorov,   "Families of plane topological mappings" , Novosibirsk  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.D. Suvorov,   "Metric theory of prime ends and boundary properties of plane mappings with bounded Dirichlet integrals" , Kiev  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  O.V. Ivanov,   G.D. Suvorov,   "Complete lattices of conformally-invariant compactifications of a domain" , Kiev  (1982)  (In Russian)</TD></TR></table>



====Comments====
Standard English references on boundary correspondence under conformal mapping and prime ends are [[#References|[a1]]]–[[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,   "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Ohtsuka,   "Dirichlet problem, extremal length and prime ends" , v. Nostrand-Reinhold  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Pommerenke,   "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR></table>