Prescribed $Q$-curvature flow on closed manifolds of even dimension

Abstract

On a closed Riemannian manifold $(M,g_0)$ of even dimension $n \geqslant 4$, the well-known prescribed $Q$-curvature problem asks whether or not there is a metric $g$ comformal to $g_0$ such that its $Q$-curvature, associated with the GJMS operator $\mathbf P_g$, is equal to a given function $f$. Letting $g = e^{2u}g_0$, this problem is equivalent to solving \[ \mathbf P_{g_0} u+Q_{g_0} = f e^{nu}, \] 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 $g_0$, \[ \frac{\partial g (t)}{\partial t}= -2\Big(Q_{g (t)} - \frac{\int_M f Q_{g(t)} d\mu_{g(t)} }{\int_M f^2 d\mu_{g(t)} }f \Big)g(t) \quad \text{ for } t >0, \] to study the problem of prescribing $Q$-curvature. Since $\int_M Q_g d\mu_g$ is conformally invariant, our analysis depends on the size of $\int_M Q_{g_0} d\mu_{g_0}$, which is assumed to satisfy \[ \int_M Q_0 d\mu_{g_0} \ne k (n-1)! \, {\rm vol}(\mathbb S^n) \quad \text{ for all } \; k = 2,3,... \] 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 theorems for prescribed $Q$-curvature problem can be derived from the convergence of the flow.