Q and Q-prime curvature in CR geometry

Abstract

The Q-curvature has been playing a central role in conformal geometry since its discovery by T. Branson. It has natural analogy in CR geometry, however, the CR Q-curvature vanishes on the boundary of a strictly pseudoconvex domain in C^{n+1} with a natural choice of contact form. This fact enables us to define a "secondary" Q-curvature, which we call Q-prime curvature (it was first introduced by J. Case and P. Yang in the case n=1). The integral of the Q-prime curvature, the total Q-prime curvature, is a CR invariant of the boundary. When n=1, it agrees with the Burns-Epstein invariant, which is a Chern-Simons type invariant in CR geometry. For all n>=1, it has non-trivial variation under the deformation of domains. Combining the variational formula with the deformation complex of CR structures, we show that the total Q-prime curvature takes local maximum at the standard CR sphere in a formal sense.