Research on geometryof eigenvalues of differential operators and submanifolds

本课题主要研究了微分算子的特征值问题的特征值和子流形的微分几何。利用多种不同的方法研究微分算子特征值问题的特征值估计和子流形的微分几何。(1) 50年前，Payne， Polya和Weinberger提出了屈曲问题特征值的一个通用不等式，这是一个非常困难的问题。提出了一种构造适当试函数的新方法，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)对于双调和算子的Dirichlet特征值问题的特征值的一般不等式的研究，我们解决了Ashbaugh在1999年提出的一个问题。(3)得到了复x射影空间和复射影空间的复子流形上拉普拉斯算子特征值的最优估计。(4)由于在n维欧几里德空间的定义域上很难求出拉普拉斯算子的第k个特征值的上界，因此几乎没有任何已知的结果。首先证明了一个代数递推公式，然后导出了拉普拉斯算子的第k个特征值的上界，它在k阶的意义上是最优的。(5)利用纳什定理，成功地构造了完全黎曼流形上拉普拉斯算子的特征值问题的试函数。利用我们的试函数，我们得到了这个特征值问题的特征值的普遍界。我们的宇宙界限是最好的。(6)研究了单位球上具有常数莫比乌斯标量曲率的紧致子流形的夹紧问题。进一步给出了这类子流形的分类。(7)给出了单位球面上常标量曲率紧致超曲面Jacobi算子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