Existence of one-sided Bryant-flow solutions

Determine whether every smooth unimodular CR 3-manifold admits a one-sided solution of Bryant’s geometric flow for sufficiently small negative time, thereby realizing the unimodular CR structure as the pseudoconcave boundary of a unimodular complex surface.

Background

The paper studies smooth strongly pseudoconvex three-dimensional CR manifolds equipped with a nowhere-zero closed section of the canonical bundle, equivalently a pseudo-Einstein contact form or a unimodular CR structure. Its main construction gives a counterexample to the Trautman conjecture by producing such a structure that is not locally CR embeddable.

Bryant introduced a geometric flow equation on a unimodular CR 3-manifold whose two-sided solvability for small positive and negative time yields a unimodular complex surface structure on the product with an interval. For the counterexamples constructed in the paper, the flow cannot have a smooth two-sided solution. The unresolved issue is whether a one-sided solution nevertheless always exists for negative time, which would realize the CR manifold as the pseudoconcave boundary of a unimodular complex surface.