Nonuniqueness of conformal metrics with constant $Q$-curvature

TL;DR

This paper demonstrates the nonuniqueness of conformal metrics with constant $Q$-curvature on various manifolds, revealing multiple solutions and bifurcations, especially on spheres and product spaces, with implications for the geometry of these metrics.

Contribution

It establishes new nonuniqueness results for constant $Q$-curvature metrics, including bifurcation phenomena and solutions with singularities, expanding understanding of conformal geometry in higher dimensions.

Findings

Multiple branches of metrics with constant $Q$-curvature bifurcate from Berger metrics.

Existence of infinitely many complete metrics conformal to product spaces like $ imes imes$ and $ imes imes$.

Solutions to singular $Q$-curvature problems on spheres with blow-up along subspheres.

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 $\geq5$. 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 $0\leq k<(n-4)/2$.