Study of Analysis on Manifolds

流形分析研究

• 批准号: 20540218
• 负责人: FURUTANI Kenro
• 金额: $ 2.75万
• 依托单位: Tokyo University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2008
• 资助国家: 日本
• 起止时间: 2008 至 2010
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-20540218/
• 关键词: sub-Riemann多様体; sub-Laplacian; 熱核; spectral zeta function; zeta-regularized determinant; ベキ零リー群; Grushin type 作用素; non-holonomic sub-bundle; Toeplitz作用素; 幾何学的量子化; non-holonomic subbundle; sub-Riemannian多様体; heat Kernel; Grushin type operator

Mainly we focused our study on the topics on analytic and geometric aspects of sub-Riemannian manifolds. Our sub-Riemannian manifolds are in the strong sense, that is、the sub-Riemannian structure is defined by a non-holonomic sub-bundle which is trivial as a vector bundle. So we have a sub-elliptic operator on such manifolds, which we call sub-Laplacian. Examples of such manifolds are three and seven dimensional spheres, or nilmanifolds and we studied their spectral zeta functions from the point of views of analytic continuation, residue calculus and the explicit determination of zeta regularized determinant. Also, we investigate a general framework to define a sub-elliptic operator from a sub-Laplacian on the total space to the base manifold through a submersion (or more specifically fiber bundle setting) and obtained a relation between their bi-characteristic flows. Especially we find a relation between sub-Riemannian geodesics on S^3 through Hopf fiber bundle S^3→CP^1 by notifying their relation with the isoperimetric aspect of curves on CP^1 and by considering double fiberings defined by left and right quaternion multiplication structure on S^3, we determine bi-characteristic flow of the spherical Grushin operator.

我们主要将研究重点放在亚军歧管的分析和几何方面的主题上。我们的亚riemannian歧管在很强的意义上，也就是说，亚riemannian的结构是由非全面的子束定义的，该子束微不足道，这是矢量束。因此，我们在这种歧管上有一个亚椭圆形操作员，我们称之为亚拉普拉斯。这种歧管的例子是三维球或尼尔曼叶夫，我们从分析延续，退休演算的观点和定期确定Zeta的明确确定的观点的观点来敲定它们的光谱Zeta函数。此外，我们研究了一个通用框架，以通过累积（或更具体地说是纤维束设置）从总空间上的亚拉普拉斯定义亚椭圆形操作员，并获得了双性表征流之间的关系。 Especially we find a relationship between sub-Riemannian geodesics on S^3 through Hopf fiber bundle S^3→CP^1 by notifying their relationship with the isoperimetric aspect of curves on CP^1 and by considering double fiberings defined by left and right quaternion multiplication structure on S^3, we determine bi-characteristic flow of the spherical Grushin operator.

项目成果

Quantization operators on Quadrics

Quadric 上的量化运算符

• 发表时间: 2008
• 期刊: Kyushu Journal of Mathematics Vol.62,No.1
• 作者: Wolfram Bauer;Kenro Furutani
• 通讯作者: Kenro Furutani

Spectral Analysis and Geometry of a sub-Laplacian and related Grushin type operators

亚拉普拉斯及相关 Grushin 型算子的谱分析和几何

• 发表时间: 2010
• 期刊: Advances in Partial Differential Equations, Birkhauser Vol.211
• 作者: Wolfram Bauer;Kenro Furutani;Chisato Iwasaki
• 通讯作者: Chisato Iwasaki

Zeta-regularized determinant of product manifolds and Kronecker's second limit formula

乘积流形的 Zeta 正则行列式和克罗内克第二极限公式

• 发表时间: 2009
• 作者: T. Sato;F. Takahashi;古谷賢朗
• 通讯作者: 古谷賢朗

Heat kernel of a sub-Laplacian and Grushin type operators

子拉普拉斯算子和 Grushin 算子的热核

• 发表时间: 2009
• 作者: M.Ishiwata;T.Ogawa;F.Takahashi;古谷賢朗
• 通讯作者: 古谷賢朗

Hilbert-Schmidt Operators and Berezin Ieerations

Hilbert-Schmidt 算子和 Berezin 迭代

• 发表时间: 2008
• 期刊: Tokyo J. Mathematis 31
• 作者: M.Grossi;F.Takahashi;W. Bauer K. Furutani;F.Takahashi;W. Bauer K. Furutani;W. Bauer K. Furutani
• 通讯作者: W. Bauer K. Furutani