1998 Fiscal Year Final Research Report Summary
Uniformization for smooth manifolds
| Project/Area Number |
09640091
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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 |
Geometry
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
FUKAKI Akito Guraduate School of Science and Engineering Tokyo Institute of Technology ; Professor, 大学院・理工学研究科 (90143247)
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| Co-Investigator(Kenkyū-buntansha) |
SHIGA Hiroshige Guraduate School of Science and Engineering Tokyo Institute of Technology ; Assi, 大学院・理工学研究科, 助教授 (10154189)
MIYAOKA Reiko Guraduate School of Science and Engineering Tokyo Institute of Technology ; Assi, 大学院・理工学研究科, 助教授 (70108182)
TSUJI Hajime Guraduate School of Science and Engineering Tokyo Institute of Technology ; Assi, 大学院・理工学研究科, 助教授 (30172000)
FUJITA Takao Guraduate School of Science and Engineering Tokyo Institute of Technology ; Prof, 大学院・理工学研究科, 教授 (40092324)
MURATA Minoru Guraduate School of Science and Engineering Tokyo Institute of Technology ; Prof, 大学院・理工学研究科, 教授 (50087079)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Kaehler-Einstein metrics / Stability / Scalar curvature |
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| Research Abstract |
1. It is an important unsolved problem to determine when compact Kaehler manifolds with positive first Chern class admit Kaehler-Einstein metric. In the case when the first Chern class is zero or negative one can always find a Kaehler-Einstein metric. In conrast to this, in the case when the first Chern class is positive there are known obstructions due to Y.Matsushima and the Head Investigator. More recently G.Tian extended the Head Invetigator's result to obtain a new invariant which can be applicable to wider class of manifolds. This new invariant is related to the stability in the sense of Mumford. Our research tried to get deeper understanding of this relationship.
2. It is an interesting question to determine which manifolds carry positive scalar curvature metrics, and has been solved by Gromov and Lawson for closed orientable manifolds of dimension greater than four. On the other hand, in the case of dimension four a new differential-topological invariant was introduced using the Seiberg-Witten equations to obstruct the existence of positive scalar curvature metrics. The Head Investigator translated a book written by John Morgan in which rudiments of this theory was explained.
3. Kaehler-Einstein manifolds are special examples of Kaehler manifolds of constant scalar curvature. There is known relationship between stable parabolic bundles over Riemann surfaces and ruled Kaehler sufaces of constant scalar curvature. We also tried to understand this relationship.
4. We obtained a simple proof of the openness of the set of extremal Kaehler casses.
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