Extensions of almost CR vector bundles

Asurumuni Ajith De Silva Waidyaratne, Purdue University

Abstract

Let $\bar\Omega$ be a smoothly bounded integrable almost complex manifold and assume that the Levi form at a given boundary point $z\sb0$ has either three positive or $n - 1$ negative eigenvalues. Suppose that (E,D) is a complex vector bundle defined over a boundary neighborhood $U\ \ \cap\ b\Omega$ about $z\sb0$. Further suppose that (E,D) is an almost CR vector bundle, i.e. the curvature $\Theta$ = D o D has no (0,2) components. Then there is a neighborhood $U\sb0$ of $z\sb0$ with $U\sb0\ \subset\subset\ U$ and a complex vector bundle $(\tilde E,\ \tilde D)$ on $U\sb0$ so that $\tilde E$ and $\tilde D$ are both extensions of E and D respectively and $(\tilde E,\tilde D)$ is an almost holomorphic vector bundle over $U\sb0$. It follows that if M is an abstract pseudoconvex CR manifold such that its Levi form at a given point $z\sb0$ has at least three positive eigenvalues, then every almost CR vector bundle is actually a CR vector bundle, i.e. every neighborhood has CR frame of sections of E.

Degree

Ph.D.

Advisors

Catlin, Purdue University.

Subject Area

Mathematics

COinS