Abstract

In this paper we study the relationship between CR-structures on 3-dimensional manifolds and Mizohata structures on 2-dimensional manifolds. Let (X, H0,1) be a 3-dimensional strictly pseudoconvex CR-manifold. Assume there is a free smooth S1-action on X, then X can be regarded as a principal S1-bundle π: XM over a smooth 2-dimensional manifold M; and assume the CR-structure H0,1 is invariant under the circle action. First, we show that the projection π*(H0,1) of H0,1 into CTM induces a Mizohata structure on M. If (X, H0,1) can be embedded into some Cn by CR-functions, then the induced Mizohata structure on M is locally integrable. Moreover, we consider when a neighborhood of π-1(p) can be embedded into C2 by CR-functions. Second, for every Mizohata structure V on M we can construct a CR circle bundle (X, H0,1) via a singular curvature form by using the theory of singular forms and singular connections over circle bundles described here. This construction can be viewed as a generalization of geometric prequantization to a degenerate situation.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 251-287
Launched on MUSE
1998-04-01
Open Access
No
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