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 $ x _ {1} = \phi ( x) $ on $ [ - 1 , 1 ] $ with $ \phi ( \pm 1 ) = \pm 1 $ and $ \phi ^ \prime ( x) > 0 $, then one can construct two analytic functions $ f _ {1} ( z , h ) $ and $ f _ {2} ( z , h ) $, where $ z = x + i y $ and $ h = \textrm{ const } $, mapping the rectangles $ | x | < 1 $, $ - h < y < 0 $ and $ | x | < 1 $, $ 0 < y < h $ univalently and conformally onto disjoint domains $ D _ {1} $ and $ D _ {2} $, respectively, in such a way that $ f _ {1} ( x , h ) = f _ {2} ( \phi ( x) , h ) $. This theorem was used (see [6]) to prove the existence of a function $ w = f ( z) $, $ f ( 0) = 0 $, $ f ( 1) = 1 $, realizing a quasi-conformal mapping of the disc $ | z | \leq 1 $ onto the disc $ | w | \leq 1 $ and possessing almost-everywhere a given characteristic $ h ( z) $, where

$$ h ( z) = \frac{w _ {\overline{z}\; } }{w _ {z} } ,\ \ | h ( z) | \leq h _ {0} < 1 , $$

and $ h ( z) $ is a measurable function defined for almost-all $ z= x+ iy $, $ | z | \leq 1 $. 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 $ x _ {1} = \phi ( x) $, 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 $ \gamma _ {1} $ on the circle $ | z | = 1 $ with end points $ a $ and $ b $, $ a \neq b $, and a function $ g ( z) $ on $ \gamma _ {1} $ with the properties: 1) at all the interior points of $ \gamma _ {1} $, $ g ( z) $ is regular and $ g ^ \prime ( z) \neq 0 $; 2) the function $ z _ {1} = g ( z) $ establishes a one-to-one mapping of $ \gamma _ {1} $ onto the complementary arc $ \gamma _ {2} $ on $ | z | = 1 $ leaving $ a $ and $ b $ invariant. Then there is a function

$$ w = F ( z) = \frac{1}{z} + a _ {1} z + \dots , $$

regular for $ | z | \leq 1 $ except at $ 0, a , b $, such that $ F ( z) = F ( g ( z) ) $ at the interior points of $ \gamma _ {1} $.

It has also been proved that there is a univalent function $ F( z) $ 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)