Advertisement

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
Article

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Baudoin, F., Bonnefont, M., Garofalo, N.: A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincare inequality. Math. Ann. 358(3–4), 833–860 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baudoin, F., Grong, E., Kuwada, K., Thalmaier, A.: SubLaplacian comparison theorems on totally geodesic Riemannian foliations. arXiv:1706.08489
  3. 3.
    Chang, S.-C., Chiu, H.-L.: On the CR analog of Obata’s theorem in a pseudohermitian \(3\)-manifold. Math. Ann. 345(1), 33–51 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chang, S.-C., Kuo, T.-J., Lai, S.-H.: Li–Yau gradient estimate and entropy formulae for the CR heat equation in a closed pseudohermitian \(3\)-manifold. J. Diff. Geom. 89, 185–216 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chang, S.-C., Han, Y.-B., Lin, C.: On the three-circle theorem and its applications in Sasakian manifolds. arXiv:1801.08858
  6. 6.
    Cheng, S.-Y., Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chow, W.-L.: Uber system von lineaaren partiellen differentialgleichungen erster orduung. Math. Ann. 117, 98–105 (1939)MathSciNetGoogle Scholar
  8. 8.
    Dong, Y.-X., Zhang, W.: Comparison theorems in pseudohermitian geometry and applications. ArXiv:1611.00539
  9. 9.
    Graham, C.R., Lee, J.M.: Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains. Duke Math. J. 57, 697–720 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Greenleaf, A.: The first eigenvalue of a Sublaplacian on a pseudohermitian manifold. Commun. Part. Differ. Equ. 10(3), 191–217 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Koranyi, A., Stanton, N.: Liouville type theorems for some complex hypoelliptic operators. J. Funct. Anal. 60, 370–377 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lee, J .M.: The Fefferman metric and pseudohermitian invariants. Trans. AMS 296, 411–429 (1986)zbMATHGoogle Scholar
  13. 13.
    Lee, J.M.: Pseudo-Einstein structure on CR manifolds. Am. J. Math. 110, 157–178 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, P.W.-Y.: Ricci curvature lower bounds on Sasakian manifolds. arXiv:1511.09381v3
  15. 15.
    Li, P.: Lecture on Harmonic Functions. UCI (2004)Google Scholar
  16. 16.
    Strichartz, R.S.: Sub-Riemannian geometry. J. Differ. Geom. 24, 221–263 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, J.-P.: Lecture Notes on Geometric Analysis. Springer, Berlin (2005)Google Scholar
  18. 18.
    Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations