CR Sub-Laplacian Comparison and Liouville-Type Theorem in a Complete Noncompact Sasakian Manifold
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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.
KeywordsCR Bochner formula Subgradient estimate Sub-Laplacian comparison theorem Liouville-type theorem Pseudohermitian Ricci Pseudohermitian torsion Ricatti inequality Sasakain manifold
Mathematics Subject ClassificationPrimary 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.
- 2.Baudoin, F., Grong, E., Kuwada, K., Thalmaier, A.: SubLaplacian comparison theorems on totally geodesic Riemannian foliations. arXiv:1706.08489
- 5.Chang, S.-C., Han, Y.-B., Lin, C.: On the three-circle theorem and its applications in Sasakian manifolds. arXiv:1801.08858
- 8.Dong, Y.-X., Zhang, W.: Comparison theorems in pseudohermitian geometry and applications. ArXiv:1611.00539
- 14.Lee, P.W.-Y.: Ricci curvature lower bounds on Sasakian manifolds. arXiv:1511.09381v3
- 15.Li, P.: Lecture on Harmonic Functions. UCI (2004)Google Scholar
- 17.Wang, J.-P.: Lecture Notes on Geometric Analysis. Springer, Berlin (2005)Google Scholar