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A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth

Part of: CR manifolds Global differential geometry

Published online by Cambridge University Press:  07 January 2019

Der-Chen Chang ,
Shu-Cheng Chang ,
Yingbo Han  and
Jingzhi Tie
Der-Chen Chang
Affiliation:
Department of Mathematics and Statistics, Georgetown University, Washington DC, 20057-0001, USA Department of Mathematics, Fu Jen Catholic University, Taipei 242, Taiwan, R.O.C. Email: chang@georgetown.edu
Shu-Cheng Chang
Affiliation:
Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan Email: scchang@math.ntu.edu.tw
Yingbo Han
Affiliation:
School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, Henan, P.R. China Email: yingbohan@163.com
Jingzhi Tie
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA, 30602-7403, USA Email: jtie@math.uga.edu
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Abstract

In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

Keywords

CR Bochner formulaheat kernelsubgradient estimateLiouvile theoremCR volume doubling propertyCR Sobolev inequalitymean value inequality

MSC classification

Primary: 32V05: CR structures, CR operators, and generalizations
Secondary: 32V20: Analysis on CR manifolds 53C56: Other complex differential geometry
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1367 - 1394
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Author D.-C. C. is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. Author S.-C. C. is partially supported by the MOST of Taiwan. (Corresponding author) Y. H. is partially supported by an NSFC grant 11201400, Nanhu Scholars Program for Young Scholars of Xinyang Normal University and the Universities Young Teachers Program of Henan Province (2016GGJS-096).

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