A Poincare-Hopf type theorem for holomorphic one forms

全纯一形式的Poincare-Hopf型定理

• 批准号: 16540086
• 负责人: ITO Toshikazu
• 金额: $ 2.24万
• 依托单位: Ryukoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540086/
• 关键词: 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

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.

我们解释主要结果。设 ω 是在圆盘 D^<2n>(1)⊂C^n 的邻域 ∪ 中定义的可积全纯单形，n【大于或等于】2 假设全纯叶状结构 F(。由 ω 定义的余维 1 的 ω) 横向于 D^<2n>(1) 的边界 S^<2n-1>(1) 通过莫比乌斯变换，我们可以假设 D^<2n>(1) 内 ω 的唯一一个奇点是原点 0。定理([6]) 如果 F(ω) 有一个叶子 L，使得 L 具有以下属性 (i) 〜(iii), 则 n=2.(i) 0∈L^^-, (ii) L 闭于 ∪＼Sing(ω), (iii) L 垂直于每个球面S^<2n-1>(r), 0<r【小于或等于】1.设 ω 为在 D^<2n>(1)⊂C^n, n【 的邻域 ∪ 上定义的全纯一式大于或等于】3 使得 Sing(ω)_∩S^<2n-1>(1)=φ. ∪.定理([7]) 如果 ω(xi)=0 且 xi 垂直于 S^<2n-1>(1)，则 ω 不可积。定理([8]) 令 X 为多项式向量场在C^n上，n【大于或等于】2且具有孤立奇点 如果由X在CP(n)上的解定义的全纯叶状结构F(X)具有双曲奇点。类型，以下条件是等效的。(i) F(X) 在 C^n 上有 n 个分隔，并且横向于球体序列 S^<2n-1>(p_j, R_j)⊂C^n 其中 <lim> ___<j→∞> R_j=+∞。(ii) X 在 C^n 上的某些仿射图中是线性的（庞加莱双曲型）。

项目成果

Existence and global stability of traveling curved fronts in the Allen-Cahn equations

• DOI: 10.1016/j.jde.2004.06.011
• 发表时间: 2005-06
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: H. Ninomiya;M. Taniguchi

Structures of positive radial solutions including singular solutions to Matsukuma's equation

正径向解的结构，包括松熊方程的奇异解

• 发表时间: 2005
• 期刊: Communication on Pure and Applied Analysis 4-4
• 作者: H.Morishita;E.Yanagida;S.Yotsutani
• 通讯作者: S.Yotsutani

On the classification of non-integrable complex distributions

不可积复分布的分类

• 发表时间: 2006
• 期刊: Indagationes Mathematicae 17-3
• 作者: T.Ito;B.Scardua
• 通讯作者: B.Scardua

Holomorphic foliation of condimension one transverse to polydiscs

多盘横向条件一的全纯叶化

• 发表时间: 2004
• 期刊: Journal for die reine undangewandte Mathomarik 575
• 作者: T.Ito;B.Scardua
• 通讯作者: B.Scardua

Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems

• DOI: 10.3934/dcds.2006.14.63
• 发表时间: 2005-10
• 期刊: Discrete and Continuous Dynamical Systems
• 影响因子: 1.1
• 作者: M. Fila;H. Ninomiya;J. Vázquez