The H-principle and Pseudoconcave CR Manifolds

TL;DR

This paper investigates the limitations of the H-principle on pseudoconcave CR manifolds, highlighting the existence of obstructions to its global validity and exploring the underlying reasons for these constraints.

Contribution

It provides initial insights into the obstructions preventing the global H-principle on pseudoconcave CR manifolds, extending understanding beyond known small-case validity.

Findings

H-principle holds locally on pseudoconcave CR manifolds

Counterexamples show failure of the H-principle globally

Identifies potential obstructions to the global H-principle

Abstract

The H-principle, which is the analogue, for CR manifolds, of the classical Hartogs principle in several complex variables, is known to be valid in the small on a pseudoconcave CR manifold of any codimension. However it fails in the large, as has been shown by the counterexample found in [HN1]. Hence there is an underlying obstruction to the global H-principle on a pseudoconcave CR manifold. The purpose of this note is to take the first steps toward a deeper understanding of this obstruction.