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

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.

Keywords

• sub-laplacian
• upper bound
• the first positive eigenvalue
• compact real hypersurfaces
• ellipsoids