| Project/Area Number |
16540086
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Ryukoku University |
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Principal Investigator |
ITO Toshikazu Ryukoku Univ., Faculty of Economics, Professor, 経済学部, 教授 (60110178)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Waichiro Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (40093314)
YOTSUTANI Shoji Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)
KOKUBU Hiroe (OKA Hiroe) Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)
NINOMIYA Hirokazu Ryukoku Univ., Faculty of Sci. and Tech., Assistant Professor, 理工学部, 助教授 (90251610)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | holomorphic one form / Poincare-Hopf type theorem / holomorphic foliation / separatrix / hyperbolic singularity / holomorphic vector field / transversity / 線形双曲ベクトル場 / 南雲型のCauchy-Kowalevskayaの定理 / 変数係数線形偏微分方程式 / 放物型偏微分方程式 / 進行波解 / 大域的漸近安定性 / 正則1-形式 / スペクトル逆問題 / Ginzburg-Landau equation / 正側葉層構造 / 複素ロジスティック方程式 / Kacの問題 / 偏微分方程式 / 反応拡散方程式 / cross-diffusion方程式 / Ginzburg Landau方程式 |
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| Research Abstract |
We explain principal results.
Let ω be an integrable holomorphic one form defined in a neighborhood ∪ of the disk D^<2n>(1)⊂C^n, n【greater than or equal】2. Assume that the holomorphic foliation F(ω) of codimension one defined by ω is transverse to the boundary S^<2n-1>(1) of D^<2n>(1). By Mobius transformation, we can suppose that the only one singular point of ω inside D^<2n>(1) is the origin 0.
Theorem([6]) If F(ω) has a leaf L such that L has the following properties (i)〜(iii), then n=2.
(i) 0∈L^^-, (ii) L is closed in ∪＼Sing(ω), (iii) L is transverse to each shpere S^<2n-1>(r), 0<r【less than or equal】1.
Let ω be a holomorphic one form defined in a neighborhood ∪ of D^<2n>(1)⊂C^n, n【greater than or equal】3 such that Sing(ω)_∩S^<2n-1>(1)=φ. Let ξ be a holomorphic vector field defined in ∪.
Theorem([7]) If ω(ξ)=0 and ξ is transverse to S^<2n-1>(1), then ω is not integrable.
Theorem([8]) Let X be a polynomial vector field on C^n, n【greater than or equal】2 with isolated singularities. If the holomorphic foliation F(X) defined by solutions of X on CP (n) has singularities of hyperbolic type, the following conditions are equivalent.
(i) F(X) has n separatrices on C^n and is transverse to a sequence of spheres S^<2n-1>(p_j, R_j)⊂C^n where <lim>___<j→∞> R_j=+∞.
(ii) X is linear (of Poincare hyperbolic type) in some affine chart on C^n.
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