Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

TL;DR

This paper introduces a new geometric condition called weak pseudoconcavity for CR manifolds, ensuring holomorphic extension of CR distributions, with applications to CR meromorphic functions and mappings, especially in higher codimension cases.

Contribution

It establishes an improved local geometric criterion for holomorphic extension on CR manifolds, extending previous concepts like essential pseudoconcavity, and provides explicit examples illustrating the new phenomena.

Findings

Germs of CR distributions are smooth and extend holomorphically under the new condition.

The condition applies to higher codimension CR manifolds, revealing phenomena absent in codimension one.

Explicit examples demonstrate the condition's applicability beyond strong pseudoconcavity.

Abstract

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1, there are additional phenomena which are invisible when d = 1.