2018 Fiscal Year Final Research Report
Research on a variational problem related to conformal maps and a variational problem of pullback of metrics
| Project/Area Number |
15K04846
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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 | Yamaguchi University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
内藤 博夫 山口大学, その他部局等(理学), 名誉教授 (10127772)
近藤 慶 山口大学, 大学院創成科学研究科, 准教授 (70736123)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | 多様体 / 変分問題 / 共形写像 / variational problem / conformal map / pullback / symphonic map / C-stationary map |
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| Outline of Final Research Achievements |
I focused on a covariant tensor for a smooth map f between Riemannian manifolds. This tensor vanishes if and only if such a map f is weakly conformal. I introduced an integral quantity and a concept of C-stationary map using this tensor and give some results on these maps. Furthermore I decomposed the quantity and obtained a functional of an integral of pullbacks of metrics. Using this functional, I introduced a concept of symphonic map, which is a counterpart of the concept of harmonic maps in a viewpoint of pullbacks of metrics. I give some results on these maps.
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| Free Research Field |
多様体上の変分問題
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| Academic Significance and Societal Importance of the Research Achievements |
2つのリーマン多様体間の写像の共形性に関して, C-stationary map という新しい概念を導入した. C-stationary map により, 2つの多様体の共形構造の違いを測るなどの応用が期待できる. C-stationary map の定義方程式は, 新しいタイプの主要項をもち, 研究が進めば, 方法論に貢献できる. さらに, この研究過程で新しい汎函数が得られ, symphonic map という概念を与えたが,「計量の pullback」という観点からは, 「harmonic map という概念の counterpart としての位置づけ」が得られる.
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