The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1676–1705 | Cite as

CR Sub-Laplacian Comparison and Liouville-Type Theorem in a Complete Noncompact Sasakian Manifold

  • Shu-Cheng Chang
  • Ting-Jung Kuo
  • Chien Lin
  • Jingzhi TieEmail author


In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e., Sasakian manifold). Second, we derive the subgradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It functions as the CR analog of Yau’s gradient estimate. As a consequence, we have the natural CR analog of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.


CR Bochner formula Subgradient estimate Sub-Laplacian comparison theorem Liouville-type theorem Pseudohermitian Ricci Pseudohermitian torsion Ricatti inequality Sasakain manifold 

Mathematics Subject Classification

Primary 32V05 32V20 Secondary 53C56 



The authors warmly thank the referee for his/her useful remarks and questions that have greatly helped to improve the paper. S.-C. Chang would like to express his gratitude to S.-T. Yau for the inspiration, C.-S. Lin for constant encouragement and supports, and J.-P. Wang for his inspiration on sub-Laplacian comparison geometry. Part of the project was done during J. Tie’s visits to Taida Institute for Mathematical Sciences (TIMS) and he would like to thank TIMS for support.


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Copyright information

© Mathematica Josephina, Inc. 2018
Corrected publication July/2018

Authors and Affiliations

  1. 1.Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS)National Taiwan UniversityTaipeiTaiwan
  2. 2.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  3. 3.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan
  4. 4.Department of MathematicsNational Tsing Hua UniversityHsinchuTaiwan
  5. 5.Department of MathematicsUniversity of GeorgiaAthensUSA

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