Nonuniqueness of Conformal Metrics With Constant Q-curvature

Abstract

Abstract We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (4th-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq 5$. 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 $\mathbb S^m\times \mathbb R^d$, $m\geq 4$, $d\geq 1$, 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 $0\leq k<(n-4)/2$.