Analysis of sub-Riemannian structures and related operators

亚黎曼结构及相关算子分析

• 批准号: 189396777
• 负责人: Professor Dr. Wolfram Bauer
• 金额: --
• 依托单位: Institut für Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/189396777?language=en
• 关键词: Analysis; sub; Riemannian; structures; related

The project will focus on the following related topics: A. Analysis of sub-Riemannian structures in the strong sense on special compact and non-compact manifolds and their homogeneous spaces. Explicit determination of the spectral data for corresponding hypo-elliptic operators and construction of their heat kernels in an integral or a geometric form. Relations between the spectral and geometric quantities. B. Analysis of the Berezin-Toeplitz quantization for generalized Segal-Bargmann spaces (cf. the description below) and for real manifolds of special type. C. We plan to study relations between sub-Riemannian structures and Berezin-Toeplitz quantization. As an application we expect to find explicit relations between classical special functions (such as theta functions, elliptic functions, hypergeometric functions • • • ) which are induced by the geometry of the underlying structures. D. We intend to study sub-Riemannian analogs of the classical Dirac operator on spinmanifolds. Since in general there is no connection canonically associated with a sub- Riemannian structure we will analyze specific examples which give additional data.

该项目将侧重于以下相关主题：A。特殊紧和非紧流形及其齐性空间上强意义下的次Riemann结构分析。显式确定相应的次椭圆算子的谱数据，并以积分或几何形式构造其热核。光谱量和几何量之间的关系。B。广义Segal-Bargmann空间的Berezin-Toeplitz量子化的分析（参见下面的描述）和特殊类型的真实的流形。C.我们计划研究亚黎曼结构和Berezin-Toeplitz量子化之间的关系。作为一个应用，我们期望找到由底层结构的几何所诱导的经典特殊函数（如θ函数、椭圆函数、超几何函数· · ·）之间的显式关系。D.我们打算研究自旋流形上经典狄拉克算子的次黎曼类似物。由于一般来说，没有连接规范与一个子黎曼结构，我们将分析具体的例子，提供额外的数据。

项目成果

Fundamental solution of a higher step Grushin type operator

更高阶Grushin型算子的基本解

• DOI: 10.1016/j.aim.2014.11.017
• 发表时间: 2015
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: W. Bauer ;K. Furutani ;C. Iwasaki
• 通讯作者: C. Iwasaki

The inverse of a parameter family of degenerate operators and applications to the Kohn-Laplacian

简并算子参数族的逆及其在 Kohn-Laplacian 中的应用

• DOI: 10.1016/j.aim.2014.12.041
• 发表时间: 2015
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: W. Bauer ;K. Furutani ;C. Iwasaki
• 通讯作者: C. Iwasaki

Spectral zeta function of the sub-Laplacian on two step nilmanifolds

两步尼尔流形上亚拉普拉斯的谱 zeta 函数

• DOI: 10.1016/j.matpur.2011.06.003
• 发表时间: 2012
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: W. Bauer ;K. Furutani ;C. Iwasaki
• 通讯作者: C. Iwasaki

Trivializable sub-Riemannian structures on spheres

球体上可平凡化的亚黎曼结构

• DOI: 10.1016/j.bulsci.2012.09.004
• 发表时间: 2013
• 期刊: Bulletin Des Sciences Mathematiques
• 影响因子: 1.3
• 作者: W. Bauer ;K. Furutani ;C. Iwasaki
• 通讯作者: C. Iwasaki