A geometric flow associated to $Q$-curvature

Abstract

Branson's $Q$-curvature is a natural fourth-order counterpart of scalar curvature in conformal geometry. In analogy with the classical recovery of the Ricci tensor from the linearized scalar curvature, Lin and Yuan introduced a symmetric $2$-tensor $J_g$ through the formal adjoint of the linearized $Q$-curvature and established its explicit decomposition in terms of the Bach tensor and auxiliary curvature terms. In this paper, we study the associated fourth-order geometric flow ${\partial }_{t}g=-2J_g$ on closed Riemannian manifolds of dimension $n\ge 4$. We establish short-time existence and uniqueness using a biharmonic DeTurck gauge and derive global and local integral smoothing estimates under curvature bounds. In dimension four, we prove long-time existence assuming a uniform Sobolev inequality and sufficiently small total $L^2$ curvature energy throughout the maximal interval of existence. Under uniform curvature and Sobolev bounds, time integrability of the $L^2$-norm of $J$ implies smooth convergence in fixed coordinates to a $J$-flat metric. Explicit Einstein and product solutions illustrate the evolution and show that small curvature energy alone does not prevent degeneration. Positive initial Yamabe constant and positive total $Q$-curvature give a uniform positive Yamabe lower bound throughout the smooth four-dimensional flow, whereas in higher dimensions total $Q$-curvature can decrease even near flat metrics. Finally, we prove that every closed four-dimensional $J$-soliton is trivial, its metric is Bach-flat with constant $Q$-curvature, and its soliton vector field is Killing.