Geometry of the Laplace operator

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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.