Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 949–963 | Cite as

The sharp upper bounds for the first positive eigenvalue of the Kohn–Laplacian on compact strictly pseudoconvex hypersurfaces

  • Song-Ying Li
  • Guijuan Lin
  • Duong Ngoc Son


We give sharp and explicit upper bounds for the first positive eigenvalue \(\lambda _1({\Box _{b}})\) of the Kohn–Laplacian on compact strictly pseudoconvex hypersurfaces in \({\mathbb {C}}^{n+1}\) in terms of their defining functions. As an application, we show that in the family of real ellipsoids, \(\lambda _1({\Box _{b}})\) has a unique maximum value at the CR sphere.


Eigenvalue Kohn–Laplacian 

Mathematics Subject Classification

32V20 32W10 


  1. 1.
    Aribi, A., Dragomir, S., El Soufi, A.: A lower bound on the spectrum of the sublaplacian. J. Geometr. Anal. 25(3), 1492–1519 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barletta, E., Dragomir, S.: On the spectrum of a strictly pseudoconvex CR manifold. Abh. Math. Semin. Univ. Hamburg 67, 33 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beals, R., Greiner, P.: Calculus on Heisenberg Manifolds, vol. 119. Princeton University Press, New Jersey (1988)zbMATHGoogle Scholar
  4. 4.
    Boutet de Monvel, L.: Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaoic-Lions-Schwartz, Expose IX (1974–1975)Google Scholar
  5. 5.
    Burns, D.: Global behavior of some tangential Cauchy–Riemann equations. Partial Differential Equations and Geometry (Proc. Conf., Park City, Utah), Marcel Dekker, New York (1979)Google Scholar
  6. 6.
    Burns, D., Epstein, C.: Embeddability for three-dimensional CR manifolds. J. Am. Math. Soc. 4, 809–840 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chanillo, S., Chiu, H.-L., Yang, P.: Embeddability for 3-dimensional Cauchy-Riemann manifolds and CR Yamabe invariants. Duke Math. J. 161(15), 2909–2921 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, S.-C., Wu, C.-T.: On the CR Obata Theorem for Kohn Laplacian in a Closed Pseudohermitian Hypersurface in \(\mathbb{C}^{n+1}\). Preprint (2012)Google Scholar
  9. 9.
    Chang, S.-C., Chiu, H.-L.: On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold. Math. Ann. 345(1), 33–51 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chiu, H.-L.: The sharp lower bound for the first positive eigenvalue of the sub-Laplacian on a pseudohermitian 3-manifold. Ann. Glob. Anal. Geom. 30(1), 81–96 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Geller, D.: The Laplacian and the Kohn Laplacian for the sphere. J. Differ. Geom. 15(3), 417–435 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Greenleaf, A.: The first eigenvalue of a sub-Laplacian on a pseudohermitian manifold. Commun. Partial Differ. Equ. 10(2), 191–217 (1985)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in the classical Domains. Transations of Mathematical Monographs, vol. 6. AMS, Providence (1963)CrossRefGoogle Scholar
  14. 14.
    Kohn, J.J.: Boundaries of complex manifolds. In: Proceedings of Conference on Complex Manifolds (Minneapolis). Springer, New York, vol. 81–94, 1965 (1964)Google Scholar
  15. 15.
    Ivanov, S., Vassilev, D.: An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence-free torsion. J. Geom. 103(3), 475–504 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, J.M.: The Fefferman metric and pseudohermitian invariants. Trans. Am. Math. Soc. 296(1), 411–429 (1986)zbMATHGoogle Scholar
  17. 17.
    Li, S.-Y., Luk, H.-S.: The Sharp lower bound for the first positive eigenvalues of sub-Laplacian on the pseudo-hermitian manifold. Proc. AMS 132, 789–798 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, S.Y., Luk, H.S.: An explicit formula for the Webster pseudo-Ricci curvature on real hypersurfaces and its application for characterizing balls in \(C^n\). Commun Anal Geom 14(4), 673–701 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, S.-Y., Son, D.N., Wang, X.-D.: A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian. Adv. Math. 281, 1285–1305 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, S.-Y., Wang, X.: An Obata-type theorem in CR geometry. J. Differ. Geom. 95(3), 483–502 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, S.-Y., Tran, M.-A.: On the CR-Obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in \({\mathbb{C}}^{n+1}\). Trans. Am. Math. Soc. 363(8), 4027–4042 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Webster, S.M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13(1), 25–41 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaIrvineUSA
  2. 2.College of Mathematics and InformaticsFujian Normal UniversityFuzhouChina
  3. 3.Science ProgramTexas A&M University at QatarEducation City, DohaQatar

Personalised recommendations