| Project/Area Number |
16H03937
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | 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)
納谷 信 名古屋大学, 多元数理科学研究科, 教授 (70222180)
塩谷 隆 東北大学, 理学研究科, 教授 (90235507)
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥17,550,000 (Direct Cost: ¥13,500,000、Indirect Cost: ¥4,050,000)
Fiscal Year 2020: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2019: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2017: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2016: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
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| Keywords | 平均曲率フロー / 最大値原理 / 特異点 / 部分多様体 / リーマン多様体 |
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| Outline of Final Research Achievements |
In this research project, by making use of the generalized maximum principle, we studied classification problems of complete self-shrinkers. Several important results are obtained. On study of λ-hypersurfaces, embedded compact λ-hypersurfaces are constructed concretely. We classified complete λ-surfaces with constant squared norm of the second fundamental form. We proved that a complete λ-hypersurface has polynomial area growth if and only if it is proper. We obtained that the lower bound growth of area of complete and non-compact λ-hypersurfaces is at least linear. Gauss-Bonnet theorem on wave front is generalized to higher dimensions in the sense of mapping degree of Gaussian map. Ends of complete and non-compact Riemannian manifolds with finite volume and negative sectional curvature are studied. Several important results are obtained. The conjecture on maximizing the first eigenvalue of Laplacian for closed surfaces with genus 2 is solved affirmatively.
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| Academic Significance and Societal Importance of the Research Achievements |
我々は独自のアイディアで新しい研究方法を開発し，完備セルフーシュリンカーの分類を研究した。λ-超曲面はセルフーシュリンカーの一般化として新しい研究課題で, 我々は完備λ-超曲面の面積増大度を研究し, 完備λ-超曲面の分類研究も行なった。 非正則点を許す曲面の幾何学は幾何学におけるとても有望な研究分野であるし, プラシアンの第1固有値の研究は幾何学及び解析学の分野で最重要な研究課題である。従って, 本研究は学術的に意義深いもので，幾何学の発展に大きく貢献することになると思われる。曲率フローは社会の様々な側面に現れるので, 学術的意義のみならず, 近い将来現実社会問題を解決に役に立つと思われる。
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