The development and fusion on the geometry of submanifolds and theory for eigenvalues of the differential operators in Riemannian manifolds

本课题利用子流形几何的研究方法与黎曼流形中有界域上微分算子特征值的研究方法的融合，我们可以构造具有良好性质的试探函数，从而得到黎曼流形中有界区域上Laplacian的Dirichlet特征值问题的特征值的一个尖锐的普适不等式，以及特征值的一个普适不等式固支板的问题。进一步，将特征值的普遍不等式与Cheng和Yang的递推公式相结合，给出了黎曼流形中有界区域上Laplacian的Dirichlet特征值问题的第k个特征值的上界.它在k阶意义下是最优的。用一种新颖的方法代替Fourier变换，得到了黎曼流形中有界区域上Laplacian算子Dirichlet特征值问题的Li-Yau型特征值下界.关于欧氏空间中有界区域上固支板问题特征值下界的研究，利用Fourier变换和区域的对称重排改进了Levine和Protter不等式.另一方面，从不同的角度研究了子流形的曲率结构和拓扑结构。给出了单位球面上常数量曲率紧致超曲面Jacobi算子第一特征值的一个最优上界。构造了单位球面中具有常数k次平均曲率的嵌入紧致超曲面。

In this project, by making use of fusion on the research methods in the geometry of submanifolds and the research methods for eigenvalues of the differential operators on bounded domains in Riemannian manifolds, we can construct trial functions with good properties such that we obtain a sharp universal inequalityfor eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domainsin Riemannian manifolds and a universal inequality for eigenvalues of the clamped plate problem. Furthermore, by combining the universal inequality for eigenvalues with therecursion formula of Cheng and Yang, we give the upper bounds for the k-th eigenvalueof the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. It is optimal in the sense of the order of k. By using a new and original method replacing the method of the Fourier transform, we obtain a Li-Yau type lower bound for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. About the study on lower bounds for eigenvalues of the clamped plate problem on bounded domains in a Euclidean space, we improve the inequality of Levine and Protter by using the Fourier transform and the symmetry rearrangement of a domain. On the other hand, we study structures of curvatures and topological structures of submanifolds according to several different view points. An optimal upper bound for the first eigenvalue of Jacobi operator of compact hypersurfaces with constant scalar curvature in the unit sphere is given. Many embedded compact hypersurfaces with constantk-th mean curvature in the unit sphere are constructed.