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The upper bound for the first positive eigenvalue of Sub-Laplacian on a compact strictly pseudoconvex hypersurface
oleh: Guijuan Lin, Sujuan Long, Qiqi Zhang
| Format: | Article |
|---|---|
| Diterbitkan: | AIMS Press 2024-08-01 |
Deskripsi
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.