The first eigenvalue for the p-Laplacian on Lagrangian submanifolds in complex space forms

Abstract

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the p-Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the p-Laplacian operator on closed orientate n-dimensional Lagrangian submanifolds in a complex space form M-n(4 epsilon) with constant holomorphic sectional curvature 4 epsilon. As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525-533] to the p-Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for epsilon = 0 and epsilon = 1, respectively.