View source for 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|Extremal metric, method of the]]; [[Strip method (analytic functions)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452001.png" /> of a given annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452002.png" /> under which the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452003.png" /> is mapped onto itself, the maximum diameter of the image of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452004.png" /> is attained if and only if the boundary component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452005.png" /> is a rectilinear segment with its centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452006.png" />. A similar result is valid for multiply-connected domains.

Out of all univalent conformal mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452007.png" /> of a given multiply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452008.png" /> with expansion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452009.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520010.png" /> at infinity and normalization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520011.png" /> at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520012.png" />, the maximum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520013.png" />, and the maximum (minimum) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520014.png" /> at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520016.png" />, are attained only on mappings that map each boundary component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520017.png" />, respectively, to an arc of a circle with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520018.png" />, or to an arc of an ellipse (hyperbola) with foci at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520020.png" />. In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520021.png" />, the disc

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520022.png" /></td> </tr></table>

is the range of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520023.png" />. Each boundary point of this disc is a value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520024.png" /> 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|quasi-conformal mapping]], and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  H. Grötzsch,   ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''82'''  (1930)  pp. 69–80</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR></table>



====Comments====
Grötzsch' theorems are [[Distortion theorems|distortion theorems]].

Cf. also [[Grötzsch principle|Grötzsch principle]].