For a closed Riemannian manifold of dimension at least three, positive Yamabe invariant implies the existence of a conformal metric with positive scalar curvature. As a higher order analogue of this result, we seek for similar characterizations for the Paneitz operator and Q-curvature in higher dimensions. For a smooth closed Riemannian manifold of dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q-curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.

There are no general existence results for prescribed negative Q-curvature problems. In the second part of my talk, I will show how to use gluing method to construct nontrivial examples of negative constant Q-curvature metrics.

Part of my talk is joint work with Matt Gursky and Fengbo Hang.