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The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces

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Abstract

Let (M,θ) be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ1 of the Kohn-Laplacian □b on (M,θ). In the present paper, we give a sharp upper bound for λ1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ1 when the pseudohermitian structure θ is volume-normalized.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Song Ying Li

  2. Department of Mathematics, Fujian Normal University, Fujian, 350108, P. R. China

    Song Ying Li

  3. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Österreich

    Duong Ngoc Son

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  1. Song Ying Li
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  2. Duong Ngoc Son
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Correspondence to Duong Ngoc Son.

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In Memory of Professor Qikeng Lu (1927–2015)

The second-named author was supported by the Austrian Science Fund FWF, Project No. I01776

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Li, S.Y., Son, D.N. The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces. Acta. Math. Sin.-English Ser. 34, 1248–1258 (2018). https://doi.org/10.1007/s10114-018-7415-0

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  • DOI: https://doi.org/10.1007/s10114-018-7415-0

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