Nonsmooth methods in geometric function theory and geometric measure theory on the Heisenberg group

摘要TysonThe建议的研究中心在微分几何，几何测度理论和几何函数理论之间的接口海森堡组和更一般的子黎曼（卡诺-Caratheodory）空间的一套问题。统一的主题是在这方面的有效工具，从非光滑度量几何的发展和应用。一系列的问题集中在几何的子流形可能应用到著名的海森堡等周猜想皮埃尔潘苏。微分几何的经典机制的次黎曼类似物最近由Garofalo等人，Pauls和Franchi等人等引入。在与Capogna和Pauls的联合工作中，PI将进一步发展这种机制，以获得对Carnot-Caratheodory子流形几何的更内在的理解。这项调查目前仅限于表面给予内在或extradership，作为水平集或参数化的高度正规的功能。PI的研究生John Maki将在他的论文中研究维数跳跃现象和弱正则函数图的特征集的大小。 第二条研究线集中于次黎曼几何函数理论和分形几何。这包括寻找有效的对称化程序，存在性，扩展和正则性问题的拟共形映射，度量规则的粗糙域（约翰和统一域，域满足一个quasiblobular增长条件），和结构的自仿射平铺。最后，与Z。M. Balogh），Cheeger-Keith的非光滑一阶微积分将与海森堡群上的奇异度量联系起来进行研究，并着眼于构建新的空间实例，在这些空间上可以开发这种微积分。亚黎曼几何是“约束运动的几何”;它为任何允许运动受到先验几何约束的物理情况提供了数学模型。 从历史上看，它的根源在于卡诺的工作热力学，但这个问题已经取得了显着的进展，远远超出了这些激励问题的中心位置，在现代非光滑几何分析，最近已经看到显着的应用在许多领域，包括机器人路径规划，卫星远程控制，数字图像重建和计算机视觉，神经生物学和金融数学。提出的研究（子流形的黎曼微分几何和等周问题）的一个方面和哺乳动物视觉皮层的功能和结构的新兴模型之间有直接的联系。非光滑技术和方法在几何分析中是必不可少的，因为经典的光滑函数和集合空间是不完备的;微分方程和变分问题的解不能得到保证，除非定义域被扩展到一个合适的大家族（非光滑）候选者（尽管事后看来，这种解的光滑性通常可以在后验中建立）。该提案包括与Capogna和Pauls联合的外联部分，涉及研究生和博士后的交叉培训，一系列会议，研讨会和暑期学校，临时文章和专著，以及为亚黎曼几何研究人员建立的在线论坛，旨在与欧洲和澳大利亚的既定研究中心一样，在这个令人兴奋的领域发展北美的存在。

Abstract TysonThe proposed research centers on a suite of problems at the interface between differential geometry, geometric measure theory and geometric function theory in the Heisenberg group and more general sub-Riemannian (Carnot-Caratheodory) spaces. The unifying theme is the development and application in this context of effective tools from nonsmooth metric geometry. One series of problems focuses on the geometry of submanifolds with possible application to the celebrated Heisenberg isoperimetry conjecture of Pierre Pansu. Sub-Riemannian analogs of the classical machinery of differential geometry have recently been introduced by Garofalo et al, Pauls, and Franchi et al, among others. In joint work with Capogna and Pauls, the PI will further develop this machinery in order to gain a more intrinsic understanding of the geometry of Carnot-Caratheodory submanifolds. This investigation is currently limited to surfaces given either intrinsically or extrinsically, as level sets of or parameterized by highly regular functions. Dimension jump phenomena and the size of the characteristic set for graphs of weakly regular functions will be investigated by the PI's graduate student John Maki in his thesis. A second line of research focuses on sub-Riemannian geometric function theory and fractal geometry. This includes the search for effective symmetrization procedures, existence, extension and regularity problems for quasiconformal maps, metric regularity of rough domains (John and uniform domains, domains satisfying a quasihyperbolic growth condition), and the structure of self-affine tilings. Finally (joint with Z. M. Balogh), the nonsmooth first-order calculus of Cheeger-Keith will be investigated in connection with exotic metrics on the Heisenberg group with an eye towards constructing new examples of spaces on which such calculus can be developed.Sub-Riemannian geometry is the "geometry of constrained motion"; it provides a mathematical model for any physical situation in which allowable motion is subject to a priori geometric constraints. Historically, its roots lie in Carnot's work on thermodynamics, but the subject has progressed significantly beyond these motivating questions to a central position in modern nonsmooth geometric analysis, and has recently seen remarkable applications in numerous areas, including robotic path planning, remote control of satellites, digital image reconstruction and computer vision, neurobiology, and the mathematics of finance. There are direct links between one aspect of the proposed research (sub-Riemannian differential geometry of submanifolds and the isoperimetric problem) and emerging models for the function and structure of the mammalian visual cortex. Nonsmooth techniques and methods are essential in geometric analysis in view of the incompleteness of classical spaces of smooth functions and sets; solutions to differential equations and variational problems cannot be guaranteed unless the domain of definition is widened to a suitably large family of (nonsmooth) candidates (although in hindsight, smoothness for such solutions can often be established a posteriori). The proposal includes an outreach component, joint with Capogna and Pauls, involving cross-training of graduate students and postdocs, a series of conferences, workshops and summer schools, expository articles and monographs, and an online forum for researchers in sub-Riemannian geometry aimed at developing a North American presence in this exciting field on par with the established centers of study in Europe and Australia.