Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces

Abstract

It is well known that the spectrum of Laplacian on a compact Riemannian manifold $M$ is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of $M$. When $M$ is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the $k+1$-th eigenvalue were given by the first $k$ eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space ${\mathbf{C}\mathbf{P}}^n\left(4\right)$ and on a compact complex hypersurface without boundary in ${\mathbf{C}\mathbf{P}}^n\left(4\right)$. We shall give an explicit estimate of the $k+1$-th eigenvalue of Laplacian on such objects by its first $k$ eigenvalues.