Smooth Trautman local-embeddability conjecture

Does a smooth three-dimensional CR manifold admit local realization in complex dimension two whenever its canonical bundle has a closed nowhere-zero section? This is the sufficiency direction of the smooth Trautman conjecture.

Why it matters and what counts as progress

Why it matters: It tests whether a differential-form condition detects local CR embeddability in the lowest relevant dimension.

What counts as progress: An embedding theorem or a smooth CR structure with the canonical section but no local embedding, with any perturbation size and regularity claims justified.

How the problem developed

1998 · Conjecture [1]
Andrzej Trautman Formulated the CR-manifold conjecture that a three-dimensional CR manifold is locally realizable in complex dimension two exactly when its canonical bundle has a closed nowhere-zero section.
1 Feb 2019 · Survey and partial result [2]
Howard Jacobowitz Reviewed the conjecture and outlined an earlier weak version of the result, keeping the smooth general assertion open.
2 Sept 2026 · Counterexample claim [3]
Sean N. Curry Reported a smooth strongly pseudoconvex nonembeddable CR 3-manifold with a nowhere-zero closed canonical-bundle section, contradicting the conjecture as stated.

AI contribution

A Smooth Counterexample to the Trautman Conjecture

Sean N. Curry

Curry constructs a nonnegative smooth, arbitrarily C²-small perturbation of a strongly pseudoconvex CR 3-structure with a nowhere-zero closed canonical-bundle section, but every local C¹ CR function has zero differential at the origin. It is therefore not locally embeddable.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull