Q-curvature flow on S4

Abstract

We study a natural counterpart of the Nirenberg problem, namely to prescribe the Q-curvature of a conformal metric on the standard S⁴ as a given function f. Our approach uses a geometric flow within the conformal class, which either leads to a solution of our problem as, in particular, in the case when f ≡ const, or otherwise induces a blow-up of the metric near some point of S⁴. Under suitable assumptions on f, also in the latter case the asymptotic behavior of the flow gives rise to existence results via Morse theory.