Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms

Abstract

This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the $\varphi$ -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the $\varphi$ -Laplacian operator on closed oriented $m$ -dimensional slant submanifolds in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}\left(\epsilon \right)$ is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the $\varphi$ -Laplacian on slant submanifold in a sphere ${\mathbb{S}}^{2n+1}$ with $\epsilon =1$ and $\varphi =2$ .