Methods and Applications of Analysis

Volume 21 (2014)

Number 3

Special issue dedicated to the 60th birthday of Stephen S.-T. Yau: Part I

Guest editors: John Erik Fornæss, Norwegian University of Science and Technology; Xiaojun Huang, Rutgers University; Song-Ying Li, University of California, Irvine; Yat Sun Poon, University of California, Riverside; Wing Shing Wong, The Chinese University of Hong Kong; and Zhouping Xin, The Institute of Mathematical Sciences, CUHK.

Flows and a tangency condition for embeddable $CR$ structures in dimension 3

Pages: 337 – 356

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n3.a4

Author

Jih-Hsin Cheng (Institute of Mathematics, Academia Sinica, Taipei, Taiwan; and National Center for Theoretical Sciences, Taipei, Taiwan)

Abstract

We study the fillability (or embeddability) of 3-dimensional $CR$ structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial $CR$ structure is fillable, then it keeps having the same property as long as the flow has a solution. We discuss the situation for the torsion flow and the Cartan flow. In the second part, we show that the above mentioned second order operator is used to express a tangency condition for the space of all fillable or embeddable $CR$ structures at one embedded in $\mathbb{C}^2$.

Keywords

$CR$ structure, fillable, embeddable, pseudohermitian structure, torsion, Tanaka-Webster curvature, Cartan flow, torsion flow

2010 Mathematics Subject Classification

Primary 32G07, 32V30. Secondary 32V05, 32V20.

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