| Project/Area Number |
24340013
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Partial Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Fukuoka University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
山田 光太郎 東京工業大学, 理学院, 教授 (10221657)
塩谷 隆 東北大学, 理学(系)研究科(研究院), 教授 (90235507)
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| Co-Investigator(Renkei-kenkyūsha) |
Futaki Akito 東京大学, 数理科学研究科, 教授 (90143247)
Koiso Miyuki 九州大学, 数理学研究院, 教授 (10178189)
Rossman Wayne 神戸大学, 理学研究科, 教授 (50284485)
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| Research Collaborator |
Li Haizhong 清華大学, 教授
Wei Guoxin 華南師範大学, 教授
Yang Hongcang 中国科学院, 教授
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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Project Status |
Completed (Fiscal Year 2016)
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| Budget Amount *help |
¥12,350,000 (Direct Cost: ¥9,500,000、Indirect Cost: ¥2,850,000)
Fiscal Year 2015: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2014: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2013: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2012: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
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| Keywords | eigenvalues of Laplacian / Riemannian manifold / minimal immersion / Alenxandrov space / Singularity / maximum principle / mean curvature flow / self-shrinker / 固有値理論 / 平均曲率フロー / 特異点 / 部分多様体 / Alexandrov空間 / ラプラス作用素 / リーマン多様体 / フロント / Ricci Soliton |
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| Outline of Final Research Achievements |
By making use of Cheng-Yang recursion formula, we give optimal estimates for lower bounds of eigenvalues of Laplacian on a bounded domain in complete Riemannian manifolds. Our method is original. According to this result, a difficult problem proposed by I. Chavel is solved. Furthermore, we find an obstruction on minimal immersions from complete Riemannian manifolds into Euclidean spaces in the view of eigenvalues of Laplacian. Geometry of fronts with singularities has been studied. Gauss-Bonnet theorem on fronts is proved. By improving the generalized maximum principle of Omori-Yau, important results on classification of complete self-shrinkers of the mean curvature flow are obtained. Eigenvalues of Laplacian on compact Alexandrov spaces are studied and important progresses are obtained.
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