Geometry of the Laplace operator

Kumura研究了带边界的非紧黎曼流形或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，他利用紧致黎曼流形的内部几何性质，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，应该准备好两个不等式，哈代和索博列夫。这些不平等很重要。实际上，例如，它们导致了Neumann热核的上界、Dirichlet热核和Green核的边界行为以及Dirichlet本征值的第一个间隙。关于非紧流形的结果，得到如下结果：一般地，在非紧黎曼流形上定义了微分算子拉普拉斯算子，它的谱与流形的几何密切相关，许多作者从不同的角度对它进行了研究。特别地，非紧完备黎曼流形的拉普拉斯的本质谱仅取决于流形的无穷大的几何。Kumura考虑了无穷远附近的曲率关于某种度量的平均，并研究了它的收敛和拉普拉斯的本质谱。他推广了Donnelly和他自己的一个结果。Kasue研究了流形的收敛与Dirichlet形式之间的关系，Sato研究了Jorgensen群，Akutagawa研究了Yamabe不变量，Okumura从整体解析的角度研究了TeichMuller空间。

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.