Glueing theorems

Theorems that establish the existence of analytic functions subject to prescribed relations on the boundary of a domain.

Lavrent'ev's glueing theorem [1]: Given any analytic function on with and , then one can construct two analytic functions and , where and , mapping the rectangles , and , univalently and conformally onto disjoint domains and , respectively, in such a way that . This theorem was used (see [6]) to prove the existence of a function , , , realizing a quasi-conformal mapping of the disc onto the disc and possessing almost-everywhere a given characteristic , where

and is a measurable function defined for almost-all , . A modified form of Lavrent'ev's theorem was also used to solve the problem of mapping a simply-connected Riemann surface conformally onto the disc [5].

Other glueing theorems (with weaker restrictions on the functions of type , see [2]) have played a major role in the theory of Riemann surfaces. Another example is as follows (see [3], [5]): Suppose one is given an arc on the circle with end points and , , and a function on with the properties: 1) at all the interior points of , is regular and ; 2) the function establishes a one-to-one mapping of onto the complementary arc on leaving and invariant. Then there is a function

regular for except at , such that at the interior points of .

It has also been proved that there is a univalent function with these properties (see [4], Chapt. 2).

References

[1]	M.A. Lavrent'ev, "Sur une classe de répresentations continues" Mat. Sb. , 42 : 4 (1935) pp. 407–424
[2]	L.I. Volkovyskii, "On the problem of the connectedness type of Riemann surfaces" Mat. Sb. , 18 : 2 (1946) pp. 185–212 (In Russian)
[3]	A.C. Schaeffer, D.C. Spencer, "Variational methods in conformal mapping" Duke Math. J. , 14 : 4 (1947) pp. 949–966
[4]	A.C. Schaeffer, D.C. Spencer, "Coefficient regions for schlicht functions" , Amer. Math. Soc. Coll. Publ. , 35 , Amer. Math. Soc. (1950)
[5]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[6]	P.P. Belinskii, "General properties of quasi-conformal mappings" , Novosibirsk (1974) pp. Chapt. 2, Par. 1 (In Russian)