Geometry of measure concentration and curvature
| Project/Area Number |
23540066
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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 |
SHIOYA Takashi 東北大学, 理学(系)研究科(研究院), 教授 (90235507)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIWARA Koji 京都大学, 大学院・理学研究科, 教授 (60229078)
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| Project Period (FY) |
2011 – 2013
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| Project Status |
Completed (Fiscal Year 2013)
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| Budget Amount *help |
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2011: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
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| Keywords | 測度の集中現象 / Levy族 / オブザーバブル距離 / 曲率次元条件 / リッチ曲率 / ラプラシアンの固有値 / ガウス測度 / 球面 / レビ族 / オブザーバブル直径 / セパレーション距離 / 相転移性質 / 無限次元空間 / リーマン多様体 / 測度距離空間 / オブザーバブル直径・距離 / ピラミッド / 測度集中 |
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| Research Abstract |
We studied the details of a geometric theory of metric measure spaces due to Gromov and wrote a book for it. We proved that if a sequence of metric measure spaces converges to a metric measure space with respect to the observable distance, then the curvature-dimension condition is stable. As an application, we give an estimate of the ratio of the k-th eigenvalue and the first eigenvalue of the Laplacian on a closed Riemannian manifold with nonnegative Ricci curvature, where the estimate depends only on k. Gromov defined a natural compactification of the space of metric measure spaces with the observable distance. We deeply considered it and introduce a new metric structure on it. We apply our metric structure to prove that an n-dimensional sphere of radius square root of n in a Euclidean space converges to an infinite-dimensional Gaussian space as n tends to infinity.
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Report
(4 results)
Research Products
(26 results)
