The Gaps of Consecutive Eigenvalues of Laplacian on Riemannian Manifolds
In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition, we also exploit the closed eigenvalue problem of Laplacian and obtain a similar optimal upper bound. As some important examples, we investigate the eigenvalues of the eigenvalue problem of the Laplacian on the unit sphere and cylinder, compact homogeneous Riemannian manifolds without boundary, connected bounded domain and compact complex hypersurface without boundary in the standard complex projective space $\mathbb{C}P^{n}(4)$ with holomorphic sectional curvature $4$, and some intrinsic estimates for the eigenvalue gap is obtained. As the author know, for the Dirichlet problem, the gap $\lambda_{k+1}-\lambda_{k}$ is bounded by the first $k$-th eigenvalues in the previous literatures. However, by a large number of numerical calculations, the author surprisingly find that the gap of the consecutive eigenvalues of the Dirichlet problem on the $n$-dimensional Euclidean space $R^{n}$ can be bounded only by the first two eigenvalues. Therefore, we venture to conjecture that all of the eigenvalues satisfy: $\lambda_{k+1}-\lambda_{k}\leq \mathfrak{S}_{i}(\Omega)(\lambda_{2}-\lambda_{1})k^{1/n}$, where $\mathfrak{S}_{i}(\Omega),i=1,2$ denote the first shape coefficient and the second shape coefficient. In particular, if we consider the second shape coefficient, then there is a close connection between this universal inequality and the famous Panye-P\'olya-Weinberger conjecture in general form. By calculating some important examples, we adduce some good evidence on the correctness of this conjecture.