Isoperimetric functions of nilpotent Lie groups

幂零李群的等周函数

• 批准号: 392328321
• 负责人: Dr. Moritz Gruber
• 金额: --
• 依托单位: Courant Institute of Mathematical Sciences
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/392328321?language=en
• 关键词: Isoperimetric; functions; nilpotent; Lie; groups

Isoperimetric functions describe the relation between the volume of subsets of a metric space and the surface area of their boundaries. A special class of them is formed by the filling functions. These measure the ratio of the volume of Lipschitz-k-cycles and the volume of Lipschitz-(k +1)-chains filling those. The growth rate of the filling functions of a metric space are quasi-isometry invariants and decode important geometric properties. For instance, they detect the rank of a symmetric space by a change in their growing behaviour.In this project we concentrate on the filling functions of nilpotent Lie groups equipped with invariant Riemannian metrics. Gromov predicted a change of their growing behaviour similar to the one for symmetric spaces. We intend to make progress in proving this conjecture and clarify its geometric meaning. For this we examine the asymptotic cones of nilpotent Lie groups and the corresponding sub-Riemannian geometry.

等距函数描述度量空间子集的体积与其边界的表面积之间的关系。其中一类特殊的函数是由填充函数组成的。这些测量了Lipschitz-k环的体积与填充这些环的Lipschitz-(k +1)链的体积之比。度量空间的填充函数的增长率是准等距不变量，它可以解码重要的几何性质。例如，它们通过生长行为的变化来检测对称空间的秩。在这个项目中，我们集中研究具有不变黎曼度量的幂零李群的填充函数。Gromov预测了它们的生长行为的变化，类似于对称空间的变化。我们打算在证明这个猜想和澄清它的几何意义方面取得进展。为此，我们研究了幂零李群的渐近锥和相应的次黎曼几何。