次黎曼流形上的拟正则映射
Quasiregular mappings on sub-Riemannian manifolds
摘要：
研究了次黎曼流形上关于Carnot-Caratheodory距离拟正则的映射，并讨论了几个相关的概念.在H-型卡诺群上，拟正则映射已经在早期用解析定义引入，但是到目前为止，在一般的次黎曼流形的情况下，还没有一个好的工作定义。因此，在本文中，我们采用了度量而不是分析的观点。作为第一个主要结果，我们证明了次黎曼透镜空间中存在非平凡一致拟正则（UQR）映射，即所有迭代的畸变有一致界的拟正则映射。在这样做的过程中，我们还获得了新的例子UQR映射的标准次黎曼球面。证明是基于一种方法，用于建设共形陷阱上使用准共形流的子黎曼领域，并适应这种方法的concentients的领域。然后可以研究由UQR映射生成的拟正则半群。在本文的第二部分中，我们遵循Tukia证明了存在一个可测的共形结构，这是不变的，在这样的半群。在这里，共形结构仅在水平分布上指定，并且回撤使用Margulis-Mostow导数（其推广了经典和Pansu导数）来定义。
英文摘要：
We study mappings on sub-Riemannian manifolds which are quasi-regular with respect to the Carnot-Caratheodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis--Mostow derivative (which generalizes the classical and Pansu derivatives).
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