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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Regularity of the Beltrami equation and $1$-quasiconformal embeddings of surfaces in $\mathbb {R}^{3}$
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by Shanshuang Yang PDF
Conform. Geom. Dyn. 13 (2009), 232-246 Request permission


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:
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 13 (2009), 232-246
  • MSC (2010): Primary 30C65; Secondary 53A05, 53A30
  • DOI:
  • MathSciNet review: 2566326