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.

等函数函数描述了度量空间的子集的体积与边界表面积之间的关系。其中的特殊类是由填充功能形成的。这些衡量了Lipschitz-k-cycles的体积与Lipschitz-（K +1） - 链链的量之比。度量空间的填充功能的增长率是准偶然不变的，并解码重要的几何特性。例如，他们通过改变其不断增长的行为的变化来检测对称空间的等级。在这个项目中，我们集中于配备有不变的Riemannian指标的Nilpotent Lie群体的填充功能。格罗莫夫（Gromov）预测，其增长行为的变化类似于对称空间的行为。我们打算在证明这种猜想并阐明其几何含义方面取得进展。为此，我们检查了nilpotent Lie基团的渐近锥和相应的亚鼠标几何形状。