A Smooth Counterexample to the Trautman Conjecture

For every neighborhood $U$ of the origin in $\mathbb{C}\times\mathbb{R}$ and every $\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\phi$, supported in $U$ with $\|\phi\|_{C^2}<\varepsilon$, such that $$T^{0,1}_\phi=\operatorname{span}_{\mathbb C}\{L_\phi\}$$ is strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.

For every neighborhood $U$ of the origin in $\mathbb{C}\times\mathbb{R}$ and every $\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\phi$, supported in $U$ with $\|\phi\|_{C^2}<\varepsilon$, such that $$T^{0,1}_\phi=\operatorname{span}_{\mathbb C}\{L_\phi\}$$ is strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.