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Unique extremality

Božin, V. and Lakic, N. and Marković, V. and Mateljević, M. (1998) Unique extremality. Journal d'Analyse Mathématique, 75 (1). pp. 299-338. ISSN 0021-7670. doi:10.1007/BF02788704.

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Extremal mappings have been one of the main topics in the theory of quasiconformal mappings since its earliest days, when Grötzsch solved the extremal problem for two rectangles. Grötzsch showed that among all quasiconformal mappings from a rectangle R_1 onto another rectangle R_2, mapping the sides of R_1 onto the corresponding sides of R_2, there exists a unique mapping with minimal dilatation (see [Gr]). Later, Teichmüller [T] generalized Grotzsch's ideas and found many extremal quasiconformal mappings, including the uniquely extremal mapping f in the set of all quasiconformal mappings homotopic to a given sense-preserving homeomorphism between two compact Riemann surfaces R and S = f(R) of genus greater than 1. In a neighborhood of all but finitely many points on the compact Riemann surface R, the extremal map f could be expressed as a conformal map followed by an affine map followed by a conformal map. The Beltrami coefficient off is of the form k|φ|φ, where 0 ≤ k < 1 and φ is an integrable holomorphic quadratic differential.

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Additional Information:© 1998 Hebrew University of Jerusalem. Received: 15 March 1998. Research of the second author supported in part by NSF grant DMS 9706769.
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Official Citation:Božin, V., Lakic, N., Marković, V. et al. J. Anal. Math. (1998) 75: 299. doi:10.1007/BF02788704
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77267
Deposited By: Tony Diaz
Deposited On:09 May 2017 21:03
Last Modified:15 Nov 2021 17:29

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