2004 Fiscal Year Final Research Report Summary
Convergence of Riemannian manifolds and spectrum of Laplacian
| Project/Area Number |
14540056
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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 | Tohoku University |
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Principal Investigator |
SHIOYA Takashi Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
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| Co-Investigator(Kenkyū-buntansha) |
KUWAE Kazuhiro Kumamoto University, Faculty of Education, Associate Professor, 教育学部, 助教授 (80243814)
FUJIWARA Koji Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
SHIOHAMA Katsuhiro Saga University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (20016059)
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| Project Period (FY) |
2002 – 2004
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| Keywords | variational convergence / Gromov-Hausdorff convergence / energy functional / CAT-space / Poincare inequality / measured metric space |
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| Research Abstract |
Let M_i→M and Y_i→Y, i=1,2,3,..., be two Gromov-Hausdorff convergent sequences of proper metric spaces, where ‘proper' means that any closed bounded set is compact. We give Radon measures on all M_i and M, and assume that the measure on M_i weakly converges to that on M. We are interested in the asymptotic behavior and convergence of maps u_i : M_i→Y_i. We introduce a concept of L^p-convergence of such u_i to a map u : M→Y,p【greater than or equal】1, and establish a theory of convergence of energy functionals E_i defined on the mapping space {u : M_i→Y_i} by generalizing Mosco's variational convergences. Mosco defined the asymptotically compactness of {E_i}, as a generalization of the Rellich compactness. The asymptotic compactness is useful to obtain the convergence of energy minimizers, i.e., harmonic maps. Under a uniform bound of the Poincare constant for E_i and some condition on the metric structure of M, we prove the asymptotic compactness of {E_i}. We say that E_i compactly converges to a functional E on {u : M→Y} if E_i Γ-converges to E and if {E_i} is asymptotically compact. We prove that the compact convergence E_i→E is equivalent to the Gromov-Hausdorff convergence of the E_i-sublevel sets to the E-sublevel sets. This gives a geometric interpretation of the compact convergence. Assume in addition that Y_i are all CAT(0)-spaces and E_i are convex and lower semi-continuous. Then, we prove that the compact convergence E_i→E is equivalent to the convergence of the corresponding resolvents, where the resolvents for E_i and E are defined by using the minimizers of the Moreau-Yosida approximation. As applications, we investigate the spectra of the Korevaar-Schoen approximating energy forms. We also obtain the compactness of the energy functionals over Riemannian manifolds under a bound of Ricci curvature.
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