$Q$-Curvature and Poincaré Metrics

This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This gives an easy proof of the result of Graham-Zworski that the log coefficient in the volume expansion of a Poincare metric is a multiple of the integral of the Q-curvature, and leads to a definition of a non-local version of the Q-curvature in odd dimensions.