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

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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.

Keywords

Eigenvalue Kohn–Laplacian 

Mathematics Subject Classification

32V20 32W10 

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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

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