On the three-circle theorem and its applications in sasakian manifolds

Shu-Cheng Chang National Taiwan University Yingbo Han Xinyang Normal University Chien Lin National Tsinghua University

Differential Geometry mathscidoc:1801.10003

This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kaehler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yauís uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.
CR three circle theorem, CR Yauís uniformization conjecture, CR holomorphic function, Sharp dimension estimate.
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@inproceedings{shu-chengon,
  title={ON THE THREE-CIRCLE THEOREM AND ITS APPLICATIONS IN SASAKIAN MANIFOLDS},
  author={Shu-Cheng Chang, Yingbo Han, and Chien Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180128152712871848913},
}
Shu-Cheng Chang, Yingbo Han, and Chien Lin. ON THE THREE-CIRCLE THEOREM AND ITS APPLICATIONS IN SASAKIAN MANIFOLDS. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180128152712871848913.
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