Lower estimates for the length of the second fundamental form of submanifolds

Abstract

In a remarkable work [35], Wei established estimates for the eigenvalues of the Laplacian on closed submanifolds $M^n$ embedded in a unit sphere $S^{n+m}$. In this study, we extend these results to the eigenvalues of the p-Laplacian. As a consequence, we provide new characterizations of the sphere $S^n$. Additionally, we derive integral inequalities in terms of the norm of the second fundamental form of M and the first non-zero eigenvalue of the p-Laplacian, thereby generalizing the results previously established by Santos and Soares [11] for hypersurfaces.