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 $x_1=\varphi (x)$ on $[-1,1]$ with $\varphi (\pm 1)=\pm 1$ and ${\varphi }^{\prime }(x)>0$, then one can construct two analytic functions $f_1(z,h)$ and $f_2(z,h)$, where $z=x+iy$ and $h=\text{ const }$, 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 $f_1(x,h)=f_2(\varphi (x),h)$. 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 $|z|\le 1$ onto the disc $|w|\le 1$ and possessing almost-everywhere a given characteristic $h(z)$, where

$$h(z)=\frac{w_{\bar{z}}}{w_z},|h(z)|\le h_0<1,$$

and $h(z)$ is a measurable function defined for almost-all $z=x+iy$, $|z|\le 1$. 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 $x_1=\varphi (x)$, 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 $b$, $a\ne b$, 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 $g^{\prime }(z)\ne 0$; 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 $|z|\le 1$ 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).

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