Riemann-Schwarz principle

Riemann–Schwarz symmetry principle

A method of extending conformal mappings and analytic functions of a complex variable, formulated by B. Riemann and justified by H.A. Schwarz in the 19th century.

The Riemann–Schwarz principle for conformal mappings is as follows. Let two domains $ D _ {1} $, $ D _ {2} $ in the complex plane $ \mathbf C $ be symmetric with respect to the real axis $ \mathbf R $, let them be non-intersecting, and let their boundaries contain a common interval $ \gamma \subset \mathbf R $, whereby $ D = D _ {1} \cup \gamma \cup D _ {2} $ is also a domain. Let $ D _ {1} ^ \star $, $ D _ {2} ^ \star $, $ \gamma ^ \star $, and $ D ^ \star $ be similarly defined. If a function $ f _ {1} $, continuous in $ D _ {1} \cup \gamma $, conformally maps $ D _ {1} $ onto $ D _ {1} ^ \star $ and if $ f _ {1} ( \gamma ) = \gamma ^ \star $, then the function $ f( z) $ equal to $ f _ {1} ( z) $ when $ z \in D _ {1} \cup \gamma $ and to $ \overline{ {f _ {1} ( z) }}\; $ when $ z \in D _ {2} $ realizes a conformal mapping of $ D $ onto $ D ^ \star $.

A more general formulation of the Riemann–Schwarz principle is obtained when $ D _ {1} $, $ D _ {2} $ and $ D _ {1} ^ \star $, $ D _ {2} ^ \star $ are domains on the Riemann sphere $ \overline{\mathbf C}\; $ that are symmetric with respect to two neighbourhoods $ C, C ^ \star \subset \overline{\mathbf C}\; $, respectively, and $ \gamma \subset C $, $ \gamma \subset C ^ \star $ are open arcs, (see Symmetry principle).

The Riemann–Schwarz principle for holomorphic functions. Let the boundary of a domain $ D \subset \mathbf C $ contain a real-analytic arc. If a function $ f $ is holomorphic in $ D $, continuous in $ D \cup \gamma $ and if its values on $ \gamma $ belong to another real-analytic arc $ \gamma ^ \star $, then $ f $ can be analytically extended to a neighbourhood of $ \gamma $.

The Riemann–Schwarz principle is used in the construction of conformal mappings of plane domains as well as in the theory of analytic extension of functions of one or several complex variables.

References

[1]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)

Comments

The Riemann–Schwarz principle is also known as the Schwarz reflection principle (cf. also Schwarz symmetry theorem). The principle can be adapted to the case of $ C ^ {k} $-arcs. Then one obtains a non-holomorphic extension. This can be used to prove smoothness up to the boundary of conformal mappings of $ C ^ {k} $-smoothly bounded domains in $ \mathbf C $. Moreover, this method has been generalized to obtain smoothness up to the boundary of biholomorphic mappings between strictly pseudo-convex, $ C ^ {k} $-smoothly bounded domains, cf. [a2], [a3].

The analogue for holomorphic functions of the Schwarz reflection principle is the famous so-called edge-of-the-wedge theorem, [a6].

References

[a1]	Z. Nehari, "Conformal mapping" , Dover, reprint (1975)
[a2]	L. Nirenberg, S. Webster, P. Yang, "Local boundary regularity of holomorphic mappings" Comm. Pure Appl. Math. , 33 (1980) pp. 305–338
[a3]	S.I. Pinchuk, S.V. Khasanov, "Asymptotically holomorphic functions and their applications" Math. USSR-Sb. , 62 : 2 (1989) pp. 541–550 Mat. Sb. , 134 (176) (1987) pp. 546–555; 576
[a4]	C. Carathéodory, "Theory of functions" , 2 , Chelsea, reprint (1954)
[a5]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
[a6]	W. Rudin, "Lectures on the edge-of-the-wedge theorem" , Amer. Math. Soc. (1971)