| Project/Area Number |
17H07092
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Single-year Grants |
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| Research Field |
Basic analysis
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| Research Institution | Kogakuin University |
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Principal Investigator |
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| Project Period (FY) |
2017-08-25 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | ベルグマン核 / 正則自己同型群 / 準円型領域 / 特殊領域 / 非有界領域 / ベルグマン写像 / 代表領域 / 等方部分群 / ベルグマン核関数 |
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| Outline of Final Research Achievements |
In this research project, we studied the isotropy subgroups of quasi-circular domains. It was known by Kaup that every element of the isotropy subgroup is a polynomial mapping. Our plan is to classify this polynomial mappings. In the previous research, we already obtained a classification for two-dimensional cases and we have a hypothesis for higher-dimensional cases. We proved that this hypothesis is true for three dimensional cases. More specifically, we showed that the Bergman mapping is a biholomorphic polynomial mapping. Moreover, forms of the Bergman mapping and its inverse are calculated explicitly.
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| Academic Significance and Societal Importance of the Research Achievements |
多変数函数論はカルタンやポアンカレの時代から続く歴史の長い研究分野である. これまで多くの研究がなされてきたが, 複素解析で考察の対象となる正則関数は局所的制約が大域的振る舞いにも影響を及ぼす. 特に対称性が高い領域では対称性の中心での制約条件が正則自己同型写像の形を線形写像の様な限られたものにするという現象も起こる. 円型領域という回転で形が変わらない領域のクラスがあるが, 本研究では対称性の度合いがこれより低い準円型領域を考察した. 特に原点を固定するという局所的条件の下で正則自己同型写像の形を分類することを目標とし, 研究計画時に立てた仮説が3次元の場合で成立していることを証明した.
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