2021 Fiscal Year Final Research Report
Studies of geometry of singularities and the eigenvalues of the Hodge-Laplacian
| Project/Area Number |
16K05117
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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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 |
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| Project Period (FY) |
2016-04-01 – 2022-03-31
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| Keywords | ホッジ・ラプラシアン / ラフ・ラプラシアン / 微分形式 / 固有値 / リーマン計量の変形 / 特異点 / L2 ストークス定理 / L2 調和形式 |
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| Outline of Final Research Achievements |
We study to reveal the geometric information of the eigenvalues of the Hodge-Lalacian on closed Riemannian manifolds, and then obtained the following results:
On any m-dimensional closed manifold, for a given degree p with 1≦p≦m-1 and a natural number k, we construct a family of Riemannian metrics with fixed volume such that the k-th eigenvalues of the Hodge-Laplacian and the rough Laplacian acting on p-forms converge to 0. In particular, in the case of m-dimensional spheres, we can choose these Riemannian metrics as thoes of non-negative sectional curvature.
This is a joint work with Colette Ann'e (Universit'e de Nantes in France).
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| Free Research Field |
微分幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
コンパクト Riemann 多様体上の Hodge-Laplacian の固有値の持つ幾何学的情報の解明は大変重要な問題である．しかし，未だその全容の解明には至っていない．解明には様々な困難があるが，まずは具体例を多く作ることが重要である．特に，幾何学的状況がつかみ易い例は重宝する．
今回，そのような具体例を構成出来たので，応用や今後の研究の方向性を与えることが出来たと考えている．
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