Extension of CR structures for noncompact CR manifolds

Jose Antonio Romero, Purdue University

Abstract

The goal of this thesis is to prove that if $(M,\ S)$ is a strictly pseudoconvex CR manifold of dimension $2n-1$ and $n\ge 2,$ then one can extend the CR structure $(M,S)$ to an integrable almost complex structure $\cal L$ on $S\sbsp{g}{+}.$ Here, $S\sbsp{g}{+} = \{(x,t)\in M\times \lbrack 0,\infty) : 0\le t\le g(x)\},$ for some smooth positive function $g(x).$ Moreover, our result leads to the following corollary: If M is a strictly pseudoconvex CR manifold of dimension $2n-1$ and $n\ge 4,$ then there exists a complex manifold $\Omega$ of dimension $2n$ so that M is holomorphically embedded in $\Omega$. D. Catlin proved these results when one has the added assumption that M is compactly contained in some larger manifold. The method of proof is to construct basis vector fields with uniform control on their coefficients. We also require that the different neighborhoods be uniformly related and that the Levi form be uniformly close to 1. This is done in chapters 2 and 3 for M. In chapter 4, we use the results of the previous two chapters to get the same uniform estimates for a thin subset of $M\times \lbrack 0,\ 1).$ We then show that the original almost complex structure is sufficiently close to being integrable, depending on the 'thinness' of the subset of $M\times \lbrack 0,\ 1).$ Then in chapter 5, we show how one can apply our results in chapter 4 to get the two main results of this thesis.

Degree

Ph.D.

Advisors

Catlin, Purdue University.

Subject Area

Mathematics

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