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 $[g]$.