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Geometry of the Laplace operator

拉普拉斯算子的几何

基本信息

批准号：
13640069
负责人：
KUMURA Hironori
金额：
$ 2.24万
依托单位：
Shizuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640069/
关键词：
Laplace operator heat kernel Green kernel spectrum Sobolev inequality

项目摘要

Kumura studied the relationship between analytic inequalities of noncompact Riemannian manifolds or compact Riemannian manifolds with boundary and its geometric information. To be concrete, he gave an intrinsic ultracontractive bound for compact Riemannian manifolds with nonconvex boundary, using their inner geometric property, by the arguments of Davies - Simon 1984. In order to do so, two inequalities, Hardy and Sobolev should be prepared. These inequalities are important. Indeed, for example, these induce an upper bound of the Neumann heat kernel, the boundary behavior of the Dirichlet heat kernel and Green kernel and the first gap of the Dirichlet eigenvalue. As for results on noncompact manifolds, the following results is obtained : generally, on noncompact Riemannian manifolds, the differential operator, Laplacian is defined, and its spectrum is closely related to the geometry of the manifolds and studied by many authors from various points of view. In particular, the essential spectrum of the Laplacian of noncompact complete Riemannian manifolds depends only on the geometry of the infinity of manifolds. Kumura considered the average of curvatures near the infinity with respect to some measure and studied its convergence and the essential spectrum of the Laplacian. He generalized a results of Donnelly and his own one.Kasue studied the relationship between convergence of manifolds and Dirichlet forms, Sato studied the Jorgensen group, Akutagawa studied the Yamabe invariant and Okumura studied Teichmuller space from the global analytic viewpoint.

Kumura研究了非紧黎曼流形和紧黎曼流形的解析不等式与其几何信息之间的关系。具体地说，他利用黎曼流形的内在几何性质，通过Davies - Simon 1984的论证，给出了具有非凸边界的紧致黎曼流形的一个内在超压缩界。为了做到这一点，两个不等式，哈代和Sobolev应准备。这些不平等很重要。事实上，例如，这些诱导的上界的诺依曼热核，边界行为的狄利克雷热核和绿色内核和第一间隙的狄利克雷特征值。关于非紧流形的结果，得到了如下结果：一般地，在非紧黎曼流形上，定义了微分算子Laplacian，它的谱与流形的几何密切相关，许多作者从不同的角度研究了它。特别地，非紧完备黎曼流形的拉普拉斯算子的本质谱仅依赖于无穷流形的几何。Kumura考虑了无穷远点曲率的平均值，并研究了它的收敛性和Laplacian的本质谱。他推广了结果唐纳利和他自己的一个。Kasue研究了收敛之间的关系的流形和狄利克雷形式，佐藤研究了乔根森群，芥川研究了Yamabe不变量和Okumura研究Teichmuller空间从全球分析的观点。