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

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

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 $D$ 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

Email:
syang@mathcs.emory.edu

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.