Nonuniqueness of conformal metrics with constant $Q$-curvature

Abstract

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 $\ge 5$. 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 ${\mathbb{S}}^m\times {\mathbb{R}}^d$, $m\ge 4$, $d\ge 1$, and ${\mathbb{S}}^m\times {\mathbb{H}}^d$, $2\le d\le m-3$; 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 $0\le k<(n-4)/2$.

LaTeX2e, 19 pages, final (revised) version. To appear in Int. Math. Res. Not. IMRN

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Cited by in corpus (4)

• Singular solutions for the constant $Q$-curvature problem
• Uniqueness of conformal metrics with constant Q-curvature on closed Einstein manifolds
• Constant Q-curvature metrics with Delaunay ends: the nondegenerate case
• Multiplicity of singular solutions to the fractional Yamabe problem on spheres