Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 981-990

CONFORMALITY OF QUASICONFORMAL MAPPINGS AT A POINT, REVISITED

Mitsuhiro Shishikura

Kyoto University, Department of Mathematics
Kyoto 606-8502, Japan; mitsu 'at' math.kyoto-u.ac.jp

Abstract. We present a new and simple proof of Teichmüller–Wittich–Belinskii's and Gutlyanskii–Martio's theorems on the conformality of quasiconformal mappings at a given point. Known proofs gave separate estimates for the radial and angular variations, but our proof unifies them using Grötzsch-type inequality for the variation of cross-ratio of four points on the Riemann sphere. We also give a sufficient condition for C^1+α-conformality.

2010 Mathematics Subject Classification: Primary 30C62.

Key words: Quasiconformal mapping.

Reference to this article: M. Shishikura: Conformality of quasiconformal mappings at a point, revisited. Ann. Acad. Sci. Fenn. Math. 43 (2018), 981-990.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4359