Grötzsch theorems

Various results on conformal and quasi-conformal mappings obtained by H. Grötzsch . He developed the strip method, which is the first general form of the method of conformal moduli (cf. Extremal metric, method of the; Strip method (analytic functions)), and used it in his systematic study of a large number of extremal problems of conformal mapping of multiply-connected (including infinitely-connected) domains, including the problems of the existence, uniqueness and geometric properties of extremal mappings. A few of the simpler Grötzsch theorems are presented below.

Of all univalent conformal mappings $ w = f ( z) $ of a given annulus $ K _ {R} = \{ {z } : {R < | z | < 1 } \} $ under which the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $ is mapped onto itself, the maximum diameter of the image of the circle $ \Gamma _ {R} = \{ {z } : {| z | = R } \} $ is attained if and only if the boundary component $ f ( \Gamma _ {R} ) $ is a rectilinear segment with its centre at the point $ w = 0 $. A similar result is valid for multiply-connected domains.

Out of all univalent conformal mappings $ w = f ( z) $ of a given multiply-connected domain $ B \ni \infty $ with expansion $ f ( z) = z + O ( 1) $ $ ( z \rightarrow \infty ) $ at infinity and normalization $ f ( z _ {0} ) = 0 $ at a given point $ z _ {0} \in B $, the maximum of $ | f ^ { \prime } ( z _ {0} ) | $, and the maximum (minimum) of $ | f ( z _ {1} ) | $ at a given point $ z _ {1} \in B $, $ z _ {1} \neq z _ {0} $, are attained only on mappings that map each boundary component of $ B $, respectively, to an arc of a circle with centre at the point $ w = 0 $, or to an arc of an ellipse (hyperbola) with foci at the points $ w = 0 $ and $ w = w ^ \prime = f ( z _ {1} ) $. In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given $ z _ {1} \in B $, the disc

$$ \left \{ {w } : { \left | w - { \frac{1}{2} } ( w ^ \prime + w ^ {\prime\prime} ) \ \right | \leq \ { \frac{1}{2} } | w ^ \prime - w ^ {\prime\prime} | \ } \right \} $$

is the range of the function $ \Phi ( f ) = \mathop{\rm ln} ( f ( z _ {1} )/z _ {1} ) $. Each boundary point of this disc is a value of $ \Phi $ on a unique mapping in the class under study with specific geometric properties.

Grötzsch was the first to propose a form of representation of a quasi-conformal mapping, and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.

References

[1a]	H. Grötzsch, "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche I" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 81 (1929) pp. 38–47
[1b]	H. Grötzsch, "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche II" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 81 (1929) pp. 217–221
[1c]	H. Grötzsch, Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 82 (1930) pp. 69–80
[1d]	H. Grötzsch, "Ueber die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche II" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 84 (1932) pp. 269–278
[2]	J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)

Comments

Grötzsch' theorems are distortion theorems.

Cf. also Grötzsch principle.