Skip to main content
Log in

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

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barletta, E., Dragomir, S.: On the spectrum of a strictly pseudoconvex CR manifold. Abh. Math. Semin. Univ. Hamburg 67, 33 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beals, R., Greiner, P.: Calculus on Heisenberg Manifolds, vol. 119. Princeton University Press, New Jersey (1988)

    MATH  Google Scholar 

  4. Boutet de Monvel, L.: Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaoic-Lions-Schwartz, Expose IX (1974–1975)

  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)

  6. Burns, D., Epstein, C.: Embeddability for three-dimensional CR manifolds. J. Am. Math. Soc. 4, 809–840 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  11. Geller, D.: The Laplacian and the Kohn Laplacian for the sphere. J. Differ. Geom. 15(3), 417–435 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greenleaf, A.: The first eigenvalue of a sub-Laplacian on a pseudohermitian manifold. Commun. Partial Differ. Equ. 10(2), 191–217 (1985)

    Article  MATH  Google Scholar 

  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)

    Book  Google Scholar 

  14. Kohn, J.J.: Boundaries of complex manifolds. In: Proceedings of Conference on Complex Manifolds (Minneapolis). Springer, New York, vol. 81–94, 1965 (1964)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, J.M.: The Fefferman metric and pseudohermitian invariants. Trans. Am. Math. Soc. 296(1), 411–429 (1986)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, S.-Y., Wang, X.: An Obata-type theorem in CR geometry. J. Differ. Geom. 95(3), 483–502 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  23. Webster, S.M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13(1), 25–41 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Duong Ngoc Son.

Additional information

The second author was partially supported by the Hu Guozan Study-Abroad Grant for graduates (China) for her visit to UC Irvine in 2015–2016 when part of this work was done. The third author was partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098. Part of this work was done while the third author visited Fujian Normal University at Fuzhou, China in July 2016 which he thanks for supports and hospitality.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, SY., Lin, G. & Son, D.N. The sharp upper bounds for the first positive eigenvalue of the Kohn–Laplacian on compact strictly pseudoconvex hypersurfaces. Math. Z. 288, 949–963 (2018). https://doi.org/10.1007/s00209-017-1922-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1922-z

Keywords

Mathematics Subject Classification

Navigation