2003 Fiscal Year Final Research Report Summary
Study on Geometry and Analysis of Conformal Manifolds and Bubbling Trees
| Project/Area Number |
14540072
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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 | Shizuoka University |
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Principal Investigator |
AKUTAGAWA Kazuo Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (80192920)
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| Co-Investigator(Kenkyū-buntansha) |
OKUYAMA Yusuke Kanazawa University, Faculty of Science, Full-time Lecturer, 理学部, 講師 (00334954)
KUMURAKI Hironori Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (30283336)
SATO Hiroki Shizuoka University, Faculty of Science, Professor, 理学部, 教授 (40022222)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Conformal Geometry / Yamabe Invariant / Scalar Curvature / Cylindrical Manifolds / Index Theorem / Mass Invariant / Inverse Mean Curvature Flow Technique / Non-linear Analysis |
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| Research Abstract |
We have studied the following:
1.Study of cylindrical and orbifold Yamabe invariants
As a generalization of the Yamabe constant/invariant of closed manifolds, we defined appropriately the orbifold Yamabe constant/invariant in terms of the cylindrical Yamabe constant/invariant.
For a cylindrical 4-manifold with positive cylindrical Yamabe invariant, we also established a method for estimating its cylindrical Yamabe invariant from above, by means of the Atiyah-Patodi-Singer L^2-index theory. Moreover, we generalized the Kobayashi inequality for Yamabe invariants to cylindrical Yamabe invariants, and studied its applications.
2.Study on the mass of compact conformal manifolds
The mass is a geometric invariant for asymptotically flat manifolds. For a compact conformal manifold (M, C) with positive Yamabe invariant, a scalar-flat, asymptotically flat manifold (M-{p},g_<AF>) is defined naturally from each initial metric g in C, where [g_<AF>]=C. Then the mass m(g ; p) is non-negative. This mass m
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(g ; p) also depends on the choice of g and p. However, if we use the Habermann-Jost's canonical metric g_<HJ> as a initial metric, then the mass m(g_<HJ>;p) is now independent of the choice of p. By using this fact, we can define the mass mass(M ; C) of the conformal manifold (M, C) as a conformal invariant. Moreover, taking the infimum of it over all conformal classes, we can also define the mass invariant mass(M) as a differential-topological invariant of M. We studied on the Kobayashi-type inequality of the mass invariant for connected manifolds.
3.Yamabe invariants of 3-manifolds
The method of inverse mean curvature flow is the central technique for the resolution of the Riemannian Penrose Conjecture in Cosmology. By using this technique, Bray-Neves determined the value of the Yamabe invariant of RP^3. This result is the first affirmative answer to the Schoen's Conjecture for the Yamabe invariant of 3-manifolds with constant curvature. We also determined the Yamabe invariant of the connected manifold RP^3 # k(S^2 x S^1), by means of the inverse mean curvature flow technique. This is also one of the open problems proposed by Bray-Neves.
For the above study, the support by the 'Grant-in-Aid for Sci. Res. (C)(2),14540072' was very important. Less
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Research Products
(12 results)
