Fefferman–Hirachi vanishing CR Q-curvature conjecture

Let $(M,J,\theta_0)$ be a closed strictly pseudoconvex CR $3$-manifold, and let $[\theta_0]$ denote its conformal contact class. A contact form $\theta$ is said to have vanishing CR $Q$-curvature when its CR $Q$-curvature satisfies $Q=0$.

Fefferman–Hirachi conjecture. There exists a contact form $\theta$ in the conformal contact class

$$[\theta_0]$$

with vanishing CR $Q$-curvature.

The conjecture asks for a zero of the CR $Q$-curvature within every conformal contact class on a closed strictly pseudoconvex CR $3$-manifold. The paper gives a partial affirmative result under additional assumptions, including vanishing first Chern class, nonnegative CR Paneitz operator with kernel consisting of CR pluriharmonic functions, and, for the stated consequence, vanishing torsion; the unrestricted conjecture remains open.