Conformality of quasiconformal mappings at a point, revisited
Mitsuhiro Shishikura
TL;DR
This paper offers a simplified proof of key theorems on the conformality of quasiconformal mappings at a point, unifying previous estimates through a Grötzsch-type inequality and providing conditions for higher regularity.
Contribution
It introduces a new, unified proof technique for conformality theorems and establishes a sufficient condition for $C^{1+eta}$-conformality.
Findings
Unified proof of conformality at a point
New Grötzsch-type inequality for cross-ratio variation
Sufficient condition for $C^{1+eta}$-conformality
Abstract
We present a new and simple proof of Teichm\"uller-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 Gr\"otzsch-type inequality for the variation of cross-ratio of four points on the Riemann sphere. We also give a sufficient condition for -conformality
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Taxonomy
TopicsAnalytic and geometric function theory · Pelvic and Acetabular Injuries · Nonlinear Partial Differential Equations