Boundary correspondence (under conformal mapping)

A property of a univalent conformal mapping $ f $ of a finitely-connected domain $ G $ onto a domain $ D $ in the $ z $- plane consisting of the fact that $ f $ can be extended to a homeomorphism between certain compactifications $ \widetilde{G} $ and $ \widetilde{D} $ of $ G $ and $ D $, respectively; that is, $ f $ induces a homeomorphism of the boundaries $ \widetilde{G} \setminus G $ and $ \widetilde{D} \setminus D $. For the ordinary (Euclidean) boundaries $ \partial G $ and $ \partial D $ of $ G $ and $ D $ this property does not always hold. For example, a conformal mapping of a disc $ K $ induces a homeomorphism of $ \partial K $ and $ \partial D $ only if $ \partial D $ 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 [1], and also [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). A theory has been developed of boundary correspondence under variable conformal mappings of simply-connected domains (see [3]).

References

[1]	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
[2]	C. Carathéodory, "Über die Begrenzung einfach zusammenhängender Gebiete" Math. Ann. , 73 (1913) pp. 323–370
[3]	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
[4]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1965–1977) (Translated from Russian)
[5]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[6]	G.D. Suvorov, "Families of plane topological mappings" , Novosibirsk (1965) (In Russian)
[7]	G.D. Suvorov, "Metric theory of prime ends and boundary properties of plane mappings with bounded Dirichlet integrals" , Kiev (1981) (In Russian)
[8]	O.V. Ivanov, G.D. Suvorov, "Complete lattices of conformally-invariant compactifications of a domain" , Kiev (1982) (In Russian)

Comments

Standard English references on boundary correspondence under conformal mapping and prime ends are [a1]–[a3].

References

[a1]	L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)
[a2]	M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand-Reinhold (1970)
[a3]	C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)