9401778 Luo The object of this research effort is to study Moebius structures (conformally flat Riemannian metrics) on 3-manifolds, and complex projective structures on Riemann surfaces. In dimension three, Kuiper's conjecture on the existence of Moebius structures on a class of circle bundles over surfaces was shown to be true. The construction of these Moebius structures is based on the Moebius groups generated by half-turns about three circles in S3. It is expected that more detailed study of these groups (the geometry of the configuration space of three circles in S3) will lead to a full generalization of Fenchel-Nielsen's work on hyperbolic surfaces to Moebius structures on Seifert 3-manifolds. It is also shown that given any number less than 2-pi, any closed 3-manifold has Moebius cone structure with cone angle the given number. It is expected that there is a singular Riemannian metric of constant scalar curvature in the conformal class represented by the Moebius structure and that the metric is unique if the scalar curvature is -1. The Hausdorff convergence of the Riemannian metrics will then introduce a topology in the space of Moebius cone structures. Many conformally flat structures on 3-manifolds will be obtained by considering the limit of these singular Riemannian metrics as the cone angle goes to 2-pi. In dimension two, it is shown that the monodromy representation locally determines the quasi-bounded complex projective structure on punctured surfaces if there are no apparent singularities. As a consequence, the Teichmueller space Tg,n supports a natural family of symplectic structures for n > 0. It is expected that these symplectic structures are dual to the Fenchel-Nielson twist vectors. The study of three-dimensional spaces, called manifolds, is a natural pursuit for creatures such as ourselves who inhabit a three-dimensional universe (neglecting time). In recent years it has become clear from work of Thurston and others that understanding 3-manifolds endowed with a locally non-Euclidean geometry known as a hyperbolic structure is an important route to the understanding of 3-manifolds in general. One should look into the multiplicity of hyperbolic structures that a manifold can support and the symmetries thereof. That is the setting for this project by Feng Luo, who rings many changes on the approach. He considers hyperbolic structures on surfaces, for which there is an extensive classical theory in terms of functions of one complex variable. His discovery of an analog of the two-generator groups of symmetries that arise in the case of surfaces has been a very fertile source of new results, including the solution of a conjecture of Kuiper. It seems likely that there is still much more to be done along these lines, that the fruits of this point of view have by no means been exhausted. ***
