Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 523-536

CROSS-RATIO DISTORTION AND DOUADY–EARLE EXTENSION: III. HOW TO CONTROL THE DILATATION NEAR THE ORIGIN

Jun Hu and Oleg Muzician

Brooklyn College of CUNY, Department of Mathematics
Brooklyn, NY 11210, U.S.A.; junhu 'at' brooklyn.cuny.edu
and Graduate Center of CUNY, Ph.D. Program in Mathematics
365 Fifth Avenue, New York, NY 10016; JHu1 'at' gc.cuny.edu

CUNY BMCC, Department of Mathematics
199 Chambers Street, New York, NY 10007, U.S.A.; OMuzician 'at' bmcc.cuny.edu

Abstract. In this paper, we study how the maximal dilatation of the Douady–Earle extension near the origin is controlled by the distortion of the boundary map on finitely many points. Consider the case of points evenly spread on the circle. We show that the maximal dilatation of the extension in a neighborhood of the origin has an upper bound only depending on the cross-ratio distortion of the boundary map on these points if and only if the number n of the points is more than 4. Furthermore, we show that the size of the neighborhood is universal for each n ≥ 5 in the sense that its size only depends on the distortion.

2010 Mathematics Subject Classification: Primary 30C62, 30F60.

Key words: Douady–Earle extension, quasisymmetric homeomorphism, cross-ratio distortion, extremal quasiconformal extension, simple earthquake map.

Reference to this article: J. Hu and O. Muzician: Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin. Ann. Acad. Sci. Fenn. Math. 44 (2019), 523-536.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4432