The upper bound for the first positive eigenvalue of Sub-Laplacian on a compact strictly pseudoconvex hypersurface

Guijuan Lin; Sujuan Long; Qiqi Zhang; Guijuan Lin; Sujuan Long; Qiqi Zhang

doi:10.3934/math.20241239

AIMS Mathematics

2024, Volume 9, Issue 9: 25376-25395. doi: 10.3934/math.20241239

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Research article

The upper bound for the first positive eigenvalue of Sub-Laplacian on a compact strictly pseudoconvex hypersurface

1.
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
2.
School of Computer and Data Science, Minjiang University, Fuzhou 350108, China
3.
School of Mathematics and Imformation Science, Nanchang Hangkong University, Nanchang 330063, China

Received: 22 July 2024 Revised: 21 August 2024 Accepted: 23 August 2024 Published: 30 August 2024
MSC : 32V05, 32V15, 32V20

Let $ (M^{2n+1}, \theta) $ be a compact strictly pseudoconvex real hypersurfaces equipped with the pseudohermitian structure $ \theta $, and $ \lambda_{1} $ be the first positive eigenvalue of sub-Laplacian $ \Delta_{b} $ on $ (M^{2n+1}, \theta) $. In this paper, we will give the upper bound of $ \lambda_{1} $ under certain conditions that "$ \text{Re}\Delta_{b}\left(\rho_j+\rho_{\bar{j}}\right)\left(2\tilde{\Delta}_{\rho}\rho_j+ |\partial \rho|_\rho^{2}n^{-1}\rho^{k}\rho_{jk}\right)\leq 0 $ (for some $ j $)" or "$ \rho_{j\bar{k}} = \delta_{jk} $" holds, and apply these results to the ellipsoids furthermore.
- sub-Laplacian,
- upper bound,
- the first positive eigenvalue,
- compact real hypersurfaces,
- ellipsoids
Citation: Guijuan Lin, Sujuan Long, Qiqi Zhang. The upper bound for the first positive eigenvalue of Sub-Laplacian on a compact strictly pseudoconvex hypersurface[J]. AIMS Mathematics, 2024, 9(9): 25376-25395. doi: 10.3934/math.20241239

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Abstract

Let $ (M^{2n+1}, \theta) $ be a compact strictly pseudoconvex real hypersurfaces equipped with the pseudohermitian structure $ \theta $, and $ \lambda_{1} $ be the first positive eigenvalue of sub-Laplacian $ \Delta_{b} $ on $ (M^{2n+1}, \theta) $. In this paper, we will give the upper bound of $ \lambda_{1} $ under certain conditions that "$ \text{Re}\Delta_{b}\left(\rho_j+\rho_{\bar{j}}\right)\left(2\tilde{\Delta}_{\rho}\rho_j+ |\partial \rho|_\rho^{2}n^{-1}\rho^{k}\rho_{jk}\right)\leq 0 $ (for some $ j $)" or "$ \rho_{j\bar{k}} = \delta_{jk} $" holds, and apply these results to the ellipsoids furthermore.

References

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