Comparison geometry of Finsler manifolds and gradient flows
| Project/Area Number |
15K04844
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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 | Osaka University (2017-2018)
Kyoto University (2015-2016) |
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Principal Investigator |
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| Research Collaborator |
KUWADA Kazumasa
PALFIA Miklos
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | フィンスラー多様体 / リッチ曲率 / 局所化 / 等周不等式 / 凸関数 / 勾配流 / 収縮性 / 分解定理 / 曲率 / 距離空間 / 固有値 / 準凸関数 / 熱流 / 関数不等式 / フィンスラー幾何 / 最適輸送理論 |
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| Outline of Final Research Achievements |
We investigated Finsler manifolds with weighted Ricci curvature bounded below, and established an isoperimetric inequality via the needle decomposition, as well as isoperimetric and functional inequalities via the nonlinear Gamma calculus. We also studied gradient flows of convex functions on metric spaces, and showed the contraction property of gradient flows in CAT(1)-spaces, and the rectifiability (finiteness of the length) of gradient curves of quasi-convex functions on CAT(0)-spaces. Furthermore, we showed that a metric measure space of positive Ricci curvature with the sharp spectral gap necessarily splits off a one-dimensional Gaussian space.
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| Academic Significance and Societal Importance of the Research Achievements |
フィンスラー多様体は、距離の非対称性や熱流の非線形性など、従来のリーマン多様体では捉えられない現象の記述を可能にする幾何学的対象として、普遍的な価値を持つ。本研究で得られた成果はフィンスラー多様体の「曲がり方」に関する研究で強力な武器となるneedle分解とガンマ解析に基づくもので、今後の発展が見込まれる。また、凸関数の勾配流は最適化理論など工学的にも重要なものであり、本研究の成果はその理論的な理解を一歩進めるものである。
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Report
(5 results)
Research Products
(42 results)
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[Presentation] Nonlinear geometric analysis on Finsler manifolds2016
Author(s)
Shin-ichi Ohta
Organizer
The First Japan-Taiwan Joint Conference on Differential Geometry & the 8th TIMS- OCAMI-WASEDA Joint International Workshop on Differential Geometry and Geometric Analysis
Place of Presentation
早稲田大学
Related Report
Int'l Joint Research / Invited
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