An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaces
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- by Gian Maria Dall’Ara and Duong Ngoc Son
- Proc. Amer. Math. Soc. 151 (2023), 123-133
- DOI: https://doi.org/10.1090/proc/16077
- Published electronically: August 12, 2022
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Abstract:
A real hypersurface in $\mathbb {C}^2$ is said to be Reinhardt if it is invariant under the standard $\mathbb {T}^2$-action on $\mathbb {C}^2$. Its CR geometry can be described in terms of the curvature function of its “generating curve”, i.e., the logarithmic image of the hypersurface in the plane $\mathbb {R}^2$. We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the $\mathbb {T}^2$-action being free). Our bound is expressed in terms of the $L^2$-norm of the curvature function of the generating curve and is attained if and only if the curve is a circle.References
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Bibliographic Information
- Gian Maria Dall’Ara
- Affiliation: Istituto Nazionale di Alta Matematica “F. Severi”, Research Unit Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
- MR Author ID: 1105845
- ORCID: 0000-0003-1168-3815
- Email: dallara@altamatematica.it
- Duong Ngoc Son
- Affiliation: Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam
- MR Author ID: 800658
- Email: son.duongngoc@phenikaa-uni.edu.vn
- Received by editor(s): October 13, 2021
- Received by editor(s) in revised form: March 9, 2022
- Published electronically: August 12, 2022
- Additional Notes: This project was begun while the first-named author was a Marie Skłodowska-Curie Research Fellow at the University of Birmingham. He was supported by the European Commission via the Marie Skłodowska-Curie Individual Fellowship “Harmonic Analysis on Real Hypersurfaces in Complex Space” (ID 841094). The second-named author was supported by the Austrian Science Fund (FWF): Projekt I4557-N
- Communicated by: Harold P. Boas
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 123-133
- MSC (2020): Primary 32V20, 32W10
- DOI: https://doi.org/10.1090/proc/16077
- MathSciNet review: 4504613