Long and Medium-Term Research: Investigation of Quasicon- formal Equivalence Classes of Spherical CR Manifolds

长期和中期研究：球面CR流形的拟形式等价类研究该奖项是在长期和中期计划下颁发的。外国卓越中心的中期研究，使美国博士后研究人员能够在国外公认的卓越中心进行为期3至12个月的联合研究，使他们能够获得其他国家独特或互补的设施、专业知识和实验条件。该奖项将支持马里兰大学帕克分校的罗伯特·米纳博士与瑞士伯尔尼大学的H·M·雷曼教授进行为期12个月的访问。Miner博士将在他对服从完整的CR流形的早期分类的基础上，研究球面CR流形的拟共形等价类。直到有限覆盖，这样的流形要么是S.N-1，一个零流形，要么是Hopf流形的Heisenberg模拟，一个同胚于S1X，S.N的流形。在具有CR结构的流形之间有一个自然的准共形同态的概念，他希望把它提炼成准共形等价。在他的论文中，Miner博士证明了同胚尼尔流形作为具有服从完整的球面CR流形的覆盖实际上是拟共形的。在维度3中，可以显式地描述映射。有了海森伯格-霍普夫流形，这个问题就更加困难了。Miner博士将探索分析这一案例的技术，并研究有关Heisenberg群的定义域的拟共形等价的相关问题。Eliashberg描述了典型地伴随于CR流形的辛流形的不变量，并证明了某些环面不是拟共形等价的，并且这些不变量可能更广泛地适用。给定一对区域，我们可以定义一个相关的拟共形容量，它在拟共形映射下可以很好地变换。由此，Reimann，Pansu等人已经能够推导出在分析准共形等价时可能有用的整体扭曲性质。Kor&gt；nyi和Reimann发展了Heisenberg群上拟共形映射的变形理论。在某些情况下，高盛和Miner通过应用这一理论产生了准共形映射，并有望将其扩展到更广泛的示例类别。这笔奖金为他的国际旅行提供了资金，并为他的家乡机构提供了250美元的固定行政津贴。

Long & Medium-Term Research: Investigation of Quasicon- formal Equivalence Classes of Spherical CR Manifolds This award is made under the Program for Long- & Medium- Term Research at Foreign Centers of Excellence, which enables U.S. postdoctoral researchers to conduct 3 to 12 months' joint research abroad at centers of proven excellence, allowing them access to unique or complementary facilities, expertise and experimental conditions in other countries. This award will support a 12-month visit by Dr. Robert Miner of the University of Maryland, College Park, with Professor H.M. Reimann, Universit>t Bern, Switzerland. Dr. Miner will investigate quasiconformal equivalence classes of spherical CR manifolds based on his earlier classification of CR manifolds with amenable holonomy. Up to finite cover, such a manifold is either S.n-1, a nilmanifold, or the Heisenberg analog of a Hopf manifold, a manifold homeomorphic to S1 X S.n. There is a natural notion of a quasiconformal homeomorphism between manifolds with CR structures, which he hopes to refine up to quasiconformal equivalence. In his dissertation, Dr. Miner showed homeomorphic nilmanifolds arising as covers of spherical CR manifolds with amenable holonomy are actually quasiconformal. In dimension three, the mappings can be explicitly described. With Heisenberg Hopf manifolds, the question is more difficult. Dr. Miner will explore techniques for analyzing this case, and for investigating related questions about quasiconformal equivalence of domains of the Heisenberg group. Eliashberg has described invariants of the symplectic manifold canonically associated to a CR manifold and has shown that certain tori are not quasiconformally equivalent, and these invariants may be more widely applicable. Given a pair of domains, one can define an associated pseudoconformal capacity, which transforms nicely under quasiconformal mappings. From this, Reimann, Pansu, and others have been able to deduce global distortion properties which may be useful in analyzing quasiconformal equivalence. Kor>nyi and Reimann have developed a theory of deformations of quasiconformal mappings on the Heisenberg group. In some cases Goldman and Miner have produced quasiconformal mappings by applying this theory, and there is hope it can be extended to a wider class of examples. The award provides funds for international travel and a flat administrative allowance of $250 for his home institution.