Abstract

On a closed Riemannian manifold (M,g0) of even dimension n≥4, the well-known prescribed Q-curvature problem asks whether there is a metric g comformal to g0 such that its Q-curvature, associated with the GJMS operator Pg, is equal to a given function f. Letting g=e2ug0, this problem is equivalent to solving Pg0u+Qg0=fenu, where Qg0 denotes the Q-curvature of g0. The primary objective of the paper is to introduce the following negative gradient flow of the time dependent metric g(t) conformal to g0, ∂g(t)∂t=−2(Qg(t)−∫MfQg(t)dμg(t)∫Mf2dμg(t)f)g(t) for t>0, to study the problem of prescribing Q-curvature. Since ∫MQgdμg is conformally invariant, our analysis depends on the size of ∫MQg0dμg0 which is assumed to satisfy ∫MQ0dμg0≠k(n−1)!vol(Sn)for all k=2,3,… The paper is twofold. First, we identify suitable conditions on f such that the gradient flow defined as above is defined to all time and convergent, as time goes to infinity, sequentially or uniformly. Second, we show that various existence results for prescribed Q-curvature problem can be derived from the convergence of the flow.