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

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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.