A note on the positivity of $Q$-curvature via the continuity method

Abstract

Suppose $(M,g)$ is a smooth, closed, $5$-dimensional Riemannian manifold with positive Yamabe invariant $Y(M,[g])>0$ and positive Yamabe-type $Q$-curvature invariant $Y_4^{\ast }(M,[g])>0$. Using the continuity method, we prove the existence of a metric in the conformal class $[g]$ with positive scalar curvature and positive $Q$-curvature, assuming an additional condition on an ``initial metric" in the class $[g]$.

12 pages. Comments are most welcome