The eigenvalue estimates of p-Laplacian of totally real submanifolds in generalized complex space forms

This approach finds new upper bounds for the first positive eigenvalue of the p-Laplacian operator using the mean and constant sectional curvatures on Riemannian manifolds. In particular, we provide several estimates for the first nonzero eigenvalue of the p-Laplacian operator on closed orientated totally real submanifolds of dimension m in a generalized complex space form $${\mathbb {M}}^n(\kappa , \varepsilon )$$ . Moreover, we generalize the Reilly-inequality of Laplacian (Reilly in Comment Math Helv 52(4):525–533, 1977) to the p-Laplacian for totally real submanifold in complex projective space and complex Euclidean space for $$\kappa =1$$ and $$\kappa =0$$ , as applications, respectively.