arXiv 1606.025892016-06math.DGmath.AP

黎曼流形上Laplacian算子的连续特征值的间隙

The Gaps of Consecutive Eigenvalues of Laplacian on Riemannian Manifolds

Lingzhong Zeng

备注：: 43pages

摘要：

本文研究完备黎曼流形上Laplacian算子的Dirichlet问题。通过构造新的试探函数，得到了序列意义下连续特征值间隙的一个精确上界，从而肯定地回答了陈正杨的一个猜想.此外，我们还利用拉普拉斯算子的闭特征值问题，得到了一个类似的最优上界。作为重要的例子，我们研究了具有全纯截面曲率$4$的标准复射影空间$\mathbb{C}P^{n}(4)$中单位球、单位柱、紧致齐次无边界黎曼流形、连通有界区域和紧致复无边界超曲面上Laplacian的特征值问题，得到了特征值间隙的一些内在估计.对于Dirichlet问题，在以往的文献中，间隙$λ_{k+1}-λ_{k}$是由第一个$k$特征值所限定的.然而，通过大量的数值计算，作者惊奇地发现$n$维欧氏空间$R^{n}$上Dirichlet问题的连续特征值的间隙仅能由前两个特征值来界定。因此，我们大胆推测所有的特征值满足：$λ_{k+1}-λ_{k}\leq \mathfrak{S}_{i}(Ω)(λ_{2}-λ_{1})k^{1/n}$，其中$\mathfrak{S}_{i}(Ω),i=1,2$表示第一形状系数和第二形状系数。特别地，如果我们考虑第二个形状系数，那么这个普遍不等式与著名的Panye-Pólya-Weinberger猜想在一般形式上有着密切的联系。通过对一些重要例子的计算，给出了证明该猜想正确性的有力证据。

英文摘要：

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 $λ_{k+1}-λ_{k}$ 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: $λ_{k+1}-λ_{k}\leq \mathfrak{S}_{i}(Ω)(λ_{2}-λ_{1})k^{1/n}$, where $\mathfrak{S}_{i}(Ω),i=1,2$ 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-Pólya-Weinberger conjecture in general form. By calculating some important examples, we adduce some good evidence on the correctness of this conjecture.

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摘要：

英文摘要：

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