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Conformal Geometry and Dynamics

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Regularity of the Beltrami equation and $1$-quasiconformal embeddings of surfaces in $\mathbb {R}^{3}$

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: A striking result in quasiconformal mapping theory states that if $D$ is a domain in $\mathbb {R} ^{n}$ (with $n\geq 3$) and $f: D\rightarrow \mathbb {R} ^{n}$ an embedding, then $f$ is $1$-QC if and only if $f$ 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 $f: \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}$ from $\mathbb {R} ^{n}$ into a higher dimensional space $\mathbb {R} ^{m}$ (with $m>n$). In this paper we focus on smooth embeddings of planar domains into $\mathbb {R} ^{3}$. In particular, we show that a $C^{1+\alpha }$-smooth surface is $1$-QC equivalent to a planar domain. We also show that a topological sphere that is $C^{1+\alpha }$-diffeomorphic to the standard sphere $\mathbb {S}^{2}$ is also $1$-QC equivalent to $\mathbb {S}^{2}$. 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

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.