Research on geometryof eigenvalues of differential operators and submanifolds

本项目主要研究微分算子的特征值问题和子流形的微分几何。本文的目的是用多种方法研究微分算子的特征值问题和子流形的微分几何的特征值估计。(1)50年前，Payne、Polya和Weinberger提出了一个关于屈曲问题特征值的普适不等式，这是一个非常困难的问题。通过提出一种构造适当试函数的新方法，我们解决了Payne、Polya和Weinberger的难题。我们的结果成为屈曲问题特征值研究中最主要的贡献之一。(2)为了研究双调和算子Dirichlet特征值问题的特征值的泛不等式，我们解决了Ashbaugh在1999年提出的一个问题。(3)复域上Laplacian特征值的最优估计 ...更多信息 x射影空间和在复射影空间的复子流形中. (4)由于在n维欧氏空间中的区域上求拉普拉斯算子的第k个特征值的上界是非常困难的，所以几乎没有任何已知的结果。首先证明了一个代数递推公式，然后给出了Laplacian的第k个特征值的一个上界，它在k阶意义下是最好的。(5)利用Nash定理，我们成功地构造了完备黎曼流形中区域上Laplacian特征值问题的试探函数。通过使用我们的试验函数，我们得到了这个特征值问题的特征值的普遍界。我们的普遍边界是最好的。(6)研究了单位球面中具有常Mobius纯量曲率的紧致子流形的拼挤问题。此外，我们还给出了这类子流形的一个分类。(7)给出了单位球面中具有常数量曲率的紧致超曲面的Jacobi算子的第一特征值的最优估计。少

In this project, we mainly investigated eigenvalues of the eigenvalue problem of differential operators and the differential geometry of submanifolds. It is our purpose to research estimates for eigenvalues of the eigenvalue problems of differential operators and the differential geometry of submanifolds by means of many different methods. (1) 50 years ago, Payne, Polya and Weinberger proposed to derive a universal inequality for eigenvalues of the buckling problem, which is a very hard problem. By initiating a new method for constructing appropriated trial functions, we solve the hard problem of Payne, Polya and Weinberger. Our results become one of the most main contributions in research for eigenvalues of the buckling problem. (2) For the research on universal inequalities for eigenvalues of a Dirichlet eigenvalue problem of the biharmonic operator, we have solved a problem proposed by Ashbaugh in 1999. (3) The optimal estimates for eigenvalues of the Laplacian on a domain in comple … More x projective spaces and in complex submanifolds of complex projective spaces are obtained. (4) Since it is very difficult to derive an upper bound for the kth eigenvalue of the Laplacian on a domain in Euclidean space of dimension n, there are no any known results about it almost. We prove an algebraic recursion formula, firstly, and then we derive an upper bound for the kth eigenvalue of the Laplacian, which is best possible in the meaning of the order of k. (5) By making use of a theorem of Nash, we construct trial functions, successfully, for the eigenvalue problem of the Laplacian on a domain in a complete Riemannian manifold. By using our trial functions, we obtain universal bounds for eigenvalues of this eigenvalue problem. Our universal bounds are best possible. (6) We study the pinching problem for compact submanifolds with constant Mobius scalar curvature in a unit sphere. Furthermore, we give a classification of this kind of submanifolds. (7) We give an optimal estimate for the first eigenvalue of Jacobi operator of compact hypersurfaces with constant scalar curvature in a unit sphere. Less