2002 Fiscal Year Final Research Report Summary
Convergence of Riemannian manifolds and Laplace operators
| Project/Area Number |
12640218
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Kanazawa University (2001-2002)
Osaka City University (2000) |
|---|
Principal Investigator |
KASUE Atsushi Kanazawa University, Department of Mathematics, Professor, 自然科学研究科, 教授 (40152657)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KATO Shin Osaka City University, Department of Mathematics, As. Prof., 大学院・理学研究科, 助教授 (10243354)
ISHIMOTO Hiroyasu Kanazawa University, Department of Mathematics, Prof., 理学部, 教授 (90019472)
NAKAO Shintaro Kanazawa University, Department of Mathematics, Prof., 理学部, 教授 (90030783)
KUMURA Hironori Shizuoka University, Department of Mathematics, As. Prof., 理学部, 助教授 (30283336)
|
|---|
| Project Period (FY) |
2000 – 2002
|
| Keywords | Riemannian manifolds / Laplace operators / energy forms / heat kernels / spectra / Dirichlet spaces / Gromov-Hausdorff convergence / spectral convergence |
|---|
| Research Abstract |
Riemannian manifolds are considered as metric spaces equipped with Riemannian distances. From this point of view, a set of compact, connected Riemannian manifolds has uniform structure defined by the Gromov-Hausdorff distance, and there are intensive activities around the convergence theory of Riemannian manifolds, which include some works from the viewpoint of spectral geometry and also diffusion processes. In 1994, we introduced a spectral distance on a set of compact, (weighted) Riemannian manifolds, using heat kernels instead of Riemannian distances, and proved some results on the spectral convergence of Riemannian manifolds. In this project, we I continued the study for further developments and proved some results as follows: (1) the energy forms "dominates" the intrinsic distances in a certain sense; the relation of domination can be expressed in terms of the energy density in the limit spaces, the volume doubling property, and the scale invariant Poincare inequality, which play important roles in our theory. (2) the energy functional not only on function spaces but also on the space of maps are able to be discussed in the same vein and the convergence of the functional can be investigated in relation to the geometric and topological properties of spaces under study. (3) Riemannian vector bundles and the energy functional on them can naturally arise as the important subject of our theory. (4) Deformations of submanifolds provide new problems in our setting.
|
Research Products
(17 results)
