Study on Geometry and Analysis of Conformal Manifolds and Bubbling Trees
Project/Area Number 
14540072

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Shizuoka University 
Principal Investigator 
AKUTAGAWA Kazuo Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (80192920)

CoInvestigator(Kenkyūbuntansha) 
OKUYAMA Yusuke Kanazawa University, Faculty of Science, Fulltime 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)

Project Period (FY) 
2002 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2003: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2002: ¥1,900,000 (Direct Cost: ¥1,900,000)

Keywords  Conformal Geometry / Yamabe Invariant / Scalar Curvature / Cylindrical Manifolds / Index Theorem / Mass Invariant / Inverse Mean Curvature Flow Technique / Nonlinear Analysis / オービフォールド / ワイル不変量 
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 4manifold with positive cylindrical Yamabe invariant, we also established a method for estimating its cylindrical Yamabe invariant from above, by means of the AtiyahPatodiSinger L^2index 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 scalarflat, 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 nonnegative. This mass m
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(g ; p) also depends on the choice of g and p. However, if we use the HabermannJost'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 differentialtopological invariant of M. We studied on the Kobayashitype inequality of the mass invariant for connected manifolds. 3.Yamabe invariants of 3manifolds 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, BrayNeves 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 3manifolds 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 BrayNeves. For the above study, the support by the 'GrantinAid for Sci. Res. (C)(2),14540072' was very important. Less

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