Nonuniqueness of conformal metrics with constant $Q$-curvature

We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifurcate from Berger metrics on spheres and complex projective spaces. These provide examples of nonisometric metrics with the same constant negative $Q$-curvature in a conformal class with negative Yamabe invariant, echoing the absence of a Maximum Principle. We also discover infinitely many complete metrics with constant $Q$-curvature conformal to $\mathbb S^m\times\mathbb R^d$, $m\geq4$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d\leq m-3$; which give infinitely many solutions to the singular constant $Q$-curvature problem on round spheres $\mathbb S^n$ blowing up along a round subsphere $\mathbb S^k$, for all $0\leq k<(n-4)/2$.