Lower bounds for the length of the second fundamental form via the first eigenvalue of the [formula omitted]-Laplacian

Abstract

A suitable Bochner type formula for a certain divergence type operator is obtained. As an application, several integral inequalities involving the squared norm of the second fundamental form of a minimal compact (with or without boundary) submanifold in the sphere and the first eigenvalue of p-Laplacian are established. Finally, lower bounds of the second fundamental form of a such minimal submanifold are proven.