Q-curvature and Poincaré Metrics

This article presents a new definition of Branson’s Q-curvature in even dimensional conformal geometry. The Q-curvature is a generalization of the scalar curvature in dimension 2: it satisfies an analogous transformation law under conformal rescalings of the metric and on conformally flat manifolds its integral is a multiple of the Euler characteristic. Our approach is motivated by the recent work [GZ]; 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 Poincaré metric associated to the conformal structure. This gives an easy proof of the result of [GZ] that the log coefficient in the volume expansion of a Poincaré 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. The Q-curvature is intimately connected with a family of conformally invariant differential operators generalizing the conformal Laplacian ∆ + n−2 4(n−1)R, for which the scalar curvature arises as the zeroth order term. (Our sign convention is such that ∆ is a positive operator.) The next operator in the family was discovered by Paneitz [Pa] and has the same principal part as ∆. Branson and Ørsted [BØ] observed that the zeroth order term of Paneitz’ operator gives rise to the quantity Q = (∆R + R − 3|Ric|2)/6 in 4 dimensions with a conformal transformation law similar to that of scalar curvature in dimension 2. This Q-curvature in dimension 4 has been the focus of tremendous activity in recent years leading to great advances in our understanding of 4 dimensional conformal geometry; see [CY] for a survey of some of this work. The full family of “conformally invariant powers of the Laplacian” was derived in [GJMS], and Branson [B] formulated the definition of the Q-curvature in general even dimensions using these operators. However, this general definition involves an analytic continuation in the dimension and the higher-dimensional Q-curvature has remained a rather mysterious object. The work [GZ] shows that the conformally invariant powers of the Laplacian and the Q-curvature arise naturally in scattering theory for Poincaré metrics associated to the conformal structure. This connection can be thought of as