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距離空間上の調和関数の研究

度量空间上调和函数的研究

基本信息

批准号：
16740034
负责人：
太田慎一
金额：
$ 2.3万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Young Scientists (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

今年度は主に,リーマン多様体におけるリッチ曲率の下限に相当する条件の,一般の測度距離空間での定式化を研究した.これは10年以上にわたって重要な問題として考えられてきたものであり,現在も活発に研究されている断面曲率を下から押さえた空間(アレクサンドロフ空間)を更に一般化した対象であると共に,Cheeger-Coldingらによるリッチ曲率を下から押さえた多様体の列の収束・崩壊理論に適切な枠組みを与えるものと期待される.具体的に得られた結果としては,Sturm及びLott-Villaniによって最近与えられたCurvature-Dimension条件よりも弱い,Measure Contraction Property(MCP)という条件を導入した.(Sturmも独立に同様の条件を導入している.)これはリーマン多様体の場合にはリッチ曲率の下限と同値であり,またリッチ曲率を下から押さえたリーマン多様体で知られている種々の性質(Bishop-Gromovの体積比較定理,Bonnet-Myersの定理など)がMCPを満たす測度距離空間に拡張される.また,アレクサンドロフ空間はMCPを満たす.更に,特に重要な結果として,MCPは測度距離空間の列の測度付Gromov-Hausdorff収束の下で保存され,これとGromovのプレコンパクト性定理を合わせると,MCPを満たす測度距離空間の族は測度付Gromov-Hausdorff位相でコンパクトであることがわかる.

The main purpose of this year is that the lower limit of the curvature of the multi-body system is quite high, and the general measurement distance is not suitable for the study of the format of the multi-body system. For more than 10 years, we have been working on critical issues for more than 10 years. Now, under the curvature of the cross-sectional area of the railway, the storage space (thermal storage space) is more general than before. Cheeger-Colding is sensitive to curvature. The theory of beam collapse is related to the theory of beam collapse. The specific results show that Sturm and Lott-Villani conditions are not valid recently and that the Curvature-Dimension condition is weak, while the Measure Contraction Property (MCP) condition is weak. (Sturm is independent and the condition is the same.) The lower limit of the curvature of the multi-body system is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as the lower limit of the curvature of the multi-body, and the curvature of the multi-body is the same as that of the multi-body, and the curvature of the multi-body is the same as that of the multi-body. The No, MCP, please. More importantly, the important results are as follows: the MCP measurement distance from the space column is used to save the Gromov-Hausdorff system, the Gromov measurement system is compatible with the performance theorem, and the MCP measurement distance is compared to the Gromov-Hausdorff phase measurement system.