Prescribed Q -curvature flow on closed manifolds of even dimension

On a closed Riemannian manifold (M,g_0) of even dimension n ≥ 4 , the well-known prescribed Q -curvature problem asks whether there is a metric g comformal to g_0 such that its Q -curvature, associated with the GJMS operator 𝐏_g , is equal to a given function f . Letting g = e^2ug_0 , this problem is equivalent to solving 𝐏_g_0 u+Q_g_0 = f e^nu, where Q_g_0 denotes the Q -curvature of g_0 . The primary objective of the paper is to introduce the following negative gradient flow of the time dependent metric g ( t ) conformal to g_0 , ∂ g (t)/∂ t= -2( Q_g (t) - ∫ _M f Q_g(t) dμ _g(t)/∫ _M f^2 dμ _g(t)f ) g(t) for t >0, to study the problem of prescribing Q -curvature. Since ∫ _M Q_g dμ _g is conformally invariant, our analysis depends on the size of ∫ _M Q_g_0 dμ _g_0 which is assumed to satisfy ∫ _M Q_0 dμ _g_0 k (n-1)! vol(𝕊^n) 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.