Global minimality of generic manifolds and holomorphic extendibility of CR functions

TL;DR

This paper proves that global minimality of a generic manifold ensures the extendability of CR functions to wedges at all points in a CR orbit, confirming a conjecture by Trepreau from 1990.

Contribution

It establishes that global minimality at a point guarantees wedge extendability of CR functions throughout the CR orbit, resolving a longstanding conjecture.

Findings

CR extendability directions are preserved along CR orbits.

Global minimality implies wedge extendability at all points in the orbit.

Confirms Trepreau's conjecture from 1990.

Abstract

Let M be a smooth generic submanifold of C^n. Tumanov showed that the direction of CR extendability parallel propagates with respect to a certain differential geometric partial connection in a quotient bundle of the normal bundle to M. M is said to be globally minimal at a point z in M if the CR orbit of z contains a neighborhood of z in M. It is shown that the vector space generated by the directions of CR-extendability of CR functions on M is preserved by the induced composed flow between two points in the same CR orbit. As an application, the main result of this paper, conjectured by J.-M. Trepreau in 1990, is established: for wedge extendability of CR functions to hold at every point in the CR-orbit of z M, it is sufficient that M be globally minimal at z.