Q Curvature on a Class of 3 Manifolds

Motivated by the strong maximum principle for Paneitz operator in dimension 5 or higher found in [GM] and the calculation of the second variation of the Green’s function pole’s value on S3 in [HY2], we study Riemannian metric on 3 manifolds with positive scalar and Q curvature. Among other things, we show it is always possible to …nd a constant Q curvature metric in the conformal class. Moreover the Green’s function is always negative away from the pole and the pole’s value vanishes if and only if the Riemannian manifold is conformal di¤eomorphic to the standard S3. Compactness of constant Q curvature metrics in a conformal class and the associated Sobolev inequality are also discussed. 1. Introduction The study of Paneitz operator and Q curvature has improved our understanding of four dimensional conformal geometry ([CGY]). On three dimensional Riemannian manifold, much less is known. However the Paneitz operator and Q curvature may contain valuable information besides those related to conformal Laplacian which is associated with the scalar curvature. These additional information may help us distinguish some conformal classes from others. The aim of this article is to understand this fourth order operator for a class of metrics on three manifolds. Recall on three manifolds, the Q curvature is given by (1.1) Q = 1 4 R 2 jRcj + 23 32 R; and the fourth order Paneitz operator is de…ned as (1.2) P' = '+ 4div [Rc (r'; ei) ei] 5 4 div (Rr') 1 2 Q': Here e1; e2; e3 is a local orthonormal frame with respect to the metric (see [B, P]). Under conformal transformation of the metric, the operator satis…es (1.3) P 4g' = Pg ( ') : Note this is similar to the conformal Laplacian operator. As a consequence we know (1.4) P 4g' d 4g = Pg ( ') d g: Here g is the measure associated with metric g. Moreover kerPg = 0, kerP 4g = 0; and under this assumption, the Green’s functions satisfy the transformation law (1.5) G 4g (p; q) = (p) 1 (q) 1 Gg (p; q) : 1 2 FENGBO HANG AND PAUL C. YANG Assume (M; g) is a smooth compact three dimensional Riemannian manifold, for u; v 2 C1 (M), we denote