  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.
