2015 Fiscal Year Final Research Report
Studies on the limit of the eigenvalues of the Hodge-Laplacian under collapsing of Riemannian manifolds
| Project/Area Number |
24740034
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Keywords | スペクトル幾何学 / ラプラシアン / 固有値 / 微分形式 / リーマン多様体の崩壊 / 特異点 |
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| Outline of Final Research Achievements |
We study the limit of the eigenvalues of the Hodge-Laplacian, when a compact Riemannian manifold collapses. In particular, we study the following collapsing of Riemannian manifolds: we take a compact Riemannian manifold obtained by gluing two compact manifolds with the same boundaries, and one part of the resulting Riemannian manifold uniformly collapses to a point. Then, we prove that all of the positive eigenvalues of the Hodge-Laplacian for collapsing Riemannian manifolds converge to all of those of a suitable Hodge-Laplacian on the limit space. We also obtain an index formula for the multiplicity of eigenvalues which converge to 0.
These results are joint works with Colette Ann'e (Universiet'e de Nantes, in France).
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| Free Research Field |
幾何学
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