2010 Fiscal Year Final Research Report
Optimal mass transport on Alexandrov spaces and Ricci curvature
| Project/Area Number |
20540058
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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 東北大学, 大学院・理学研究科, 教授 (90235507)
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| Co-Investigator(Renkei-kenkyūsha) |
KUWAE Kazuhiro 熊本大学, 大学院・自然科学研究科, 教授 (80243814)
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| Project Period (FY) |
2008 – 2010
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| Keywords | 幾何学 / 測度距離空間 / 分割定理 / 体積の比較 / 測度集中 |
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| Research Abstract |
We prove that given an Alexandrov space and a positive Radon measure on it, if the measure satisfies a comparison condition of Bishop-Gromov type and if the space contains a straight line, then the space is homeomorphic to the direct product of some space and the real line. This is a generalization of the Cheeger-Gromall splitting theorem.
As another result, given a sequence of closed Riemannian manifolds of nonnegative Ricci curvature and with a uniform upper bound of diameter, if the k-th eigenvalue of the Laplacian of the manifold in the sequence is divergent to infinity, then the first eigenvalue is also divergent and the measure concentration happens.
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