2002 Fiscal Year Final Research Report Summary
Geometric structure of complete Riemannian manifolds and the scalar curvature equation
| Project/Area Number |
11640209
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka City University |
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Principal Investigator |
KATO Shin Osaka City University, Graduate school of science, associated professor, 大学院・理学研究科, 助教授 (10243354)
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| Co-Investigator(Kenkyū-buntansha) |
KASUE Atsushi Kanazawa University, Department of science, professor, 理学部, 教授 (40152657)
HASHIMOTO Yoshitake Osaka City University, Graduate school of science, associated professor, 大学院・理学研究科, 助教授 (20271182)
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| Project Period (FY) |
1999 – 2002
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| Keywords | scalar curvature / conformal deformation |
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| Research Abstract |
This project is a reserch on the scalar curvature equation that is an analytic formulation of the problem "Which kind of smooth function on a Riemannian manifold can be realized as the scalar curvature of a Riemannian metric which is pointwise conformal to the given metric ?" In this project, we deal with the case of noncompact complete Riemannian manifolds.
To take a bload view of the problem, as a reserch of geometric structure of complete Riemannian manifolds, we held, in 1999-2002, a series of meetings on the variational problems, e.g. harmonic maps, spectral geometry and the collapse of Riemannian manifolds, the graph theory, the motion of elastic curves etc., each of which has something in common with the scalar curvature equation.
In 2000, the head investigator wrote a survey on the scalar curvature equation on open Riemannian manifolds.
In 2000-2002, he also investigated on the separating phenomenon which occurs with concentration of curvature, which we can regard as a kind of bubble, and got some estimates on the separation of a Riemannian manifold by a new invariant called relative weight of end-pairs, in the model case of minimal surfaces.
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