Regularity of the Beltrami equation and -quasiconformal embeddings of surfaces in

Author:
Shanshuang Yang

Journal:
Conform. Geom. Dyn. **13** (2009), 232-246

MSC (2010):
Primary 30C65; Secondary 53A05, 53A30

Published electronically:
November 16, 2009

MathSciNet review:
2566326

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Abstract | References | Similar Articles | Additional Information

Abstract: A striking result in quasiconformal mapping theory states that if is a domain in (with ) and an embedding, then is -QC if and only if is a Möbius transformation. This result has profound impact in quasiconformal analysis and differential geometry. This project reflects part of our effort to extend this type of rigidity results to embeddings from into a higher dimensional space (with ). In this paper we focus on smooth embeddings of planar domains into . In particular, we show that a -smooth surface is -QC equivalent to a planar domain. We also show that a topological sphere that is -diffeomorphic to the standard sphere is also -QC equivalent to . Along the way, a regularity result is established for solutions of the Beltrami equation with degenerate coefficient, which is used in this paper and has its own interest.

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Additional Information

**Shanshuang Yang**

Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

Email:
syang@mathcs.emory.edu

DOI:
http://dx.doi.org/10.1090/S1088-4173-09-00200-8

Keywords:
Quasiconformal map,
conformal map,
Beltrami equation,
regularity

Received by editor(s):
August 19, 2009

Published electronically:
November 16, 2009

Additional Notes:
This research was supported in part by the University Research Committee of Emory University.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.