Eigenvalue inequalities for the p-Laplacian operator on C-totally real submanifolds in Sasakian space forms

ABSTRACT

The main objective of this paper is to provide various estimates of the first eigenvalue of the p-Laplacian operator on a closed oriented n-dimensional C-totally real submanifold ${\mathrm{\Sigma }}^n$ in a simply connected Sasakian space form ${\tilde{\mathbf{M}}}^{2m+1}\left(\kappa \right)$ with a constant ϕ-sectional curvature κ. As applications, we generalize the Reilly-type inequality for the Laplacian [Chen and Wei. Reilly-type inequalities for p-Laplacian on submanifolds in space forms. Nonlinear Anal. 2019;84:210–217; Du and Mao. Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds. Front Math China. 2015;10(3):583–594] to the p-Laplacian for C-totally real submanifold in a sphere ${\mathbb{S}}^{2m+1}$, for a constant curvature $\kappa =1$ and p = 2.

Keywords:

• Reilly-type inequality
• p-Laplacian operator
• eigenvalue estimates
• simply connected Sasakian space forms

2010 Mathematics Subject Classifications:

• 58C40
• 53C42
• 35P15

Acknowledgments

The authors would like to express their gratitude to Deanship of Scientific Research at King Khalid University, Saudi Arabia for providing funding research groups under the research grant number R. G. P. 2/71/41. Akram Ali would like to thank to the Departamento de Matematica, Universidade Federal do Ceara, Brazil, where a part of this work was carried out. He is grateful to Professors Ernani Ribeiro Jr, for the warm hospitality and their constant encouragement.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors would like to express their gratitude to Deanship of Scientific Research at King Khalid University, Saudi Arabia for providing funding research groups under the research grant number R. G. P. 2/71/41.