CR circle bundles and Mizohata structures

Anbo Le, Purdue University

Abstract

In this thesis we study the relation between CR-structures on three-dimensional manifolds and Mizohata structures on two-dimensional manifolds. Let $(X, H\sp{0,1})$ be a three-dimensional strictly pseudoconvex CR-manifold. Assume there is a free smooth $S\sp1$-action on X, then X can be regarded as a principal $S\sp1$-bundle $\pi:X\to M$ over a smooth two-dimensional manifold M; and assume the CR-structure $H\sp{0,1}$ is invariant under the circle action. First, I showed that the projection $\pi\sb{\*}(H\sp{0,1})$ of $H\sp{0,1}$ into CTM induces a Mizohata structure on M. Conversely, for every Mizohata structure V on M we can construct a CR circle bundle $(X, H\sp{0,1})$ via a singular curvature form with singularity $\Sigma,$ by using the theory of singular forms and singular connections over circle bundles described in the thesis. Second, if $(X, H\sp{0,1})$ can be embedded into some $C\sp{n}$ by CR functions, then the induced Mizohata structure on M is locally integrable. Moreover if we look at a trivialization $\pi\sp{-1}(U)\to U\times S\sp1$ where U is a neighborhood of p, then $H\sp{0,1}$ is generated locally by $L+i\alpha\sigma$ where $\alpha$ is a complex-valued function on U; and the differential equation $L(f)=\alpha$ has a solution if $\pi\sp{-1}(U)$ can be embedded into some $C\sp{n}$ by CR functions. Conversely, the local integrability of the Mizohata structure and the solvability of the differential equation $L(f)=\alpha$ are sufficient to ensure the embedding of a neighborhood of $\{p\}\times S\sp1$ into $C\sp2$ by CR functions.

Degree

Ph.D.

Advisors

Lempert, Purdue University.

Subject Area

Mathematics

COinS