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