Nonuniqueness results for constant sixth order $Q$-curvature metrics on spheres with higher dimensional singularities

Abstract

We prove nonuniqueness results for constant sixth order $Q$-metrics on complete locally conformally flat $n$-dimensional Riemannian manifolds with $n\geqslant 7$. More precisely, assuming a positive Green function exists for the sixth order GJMS operator, our objective is two-fold. First, we use a classical bifurcation technique to prove that there exists infinitely many constant $Q$-curvature metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$. As a by-product, we find the sixth order Yamabe invariant on this product manifold can be arbitrarily close to that of the round dimensional sphere, generalizing a result of Schoen about the classical Yamabe invariant. Second, when the underlying manifold is noncompact, we apply a bifurcation technique on Riemannian covering to construct infinitely many complete metrics with constant sixth order $Q$-curvature conformal to $\mathbb{S}^{n_1} \times \mathbb{R}^{n_2}$ or $\mathbb{S}^{n_1} \times \mathbb{H}^{n_2}$, where $n_1+n_2\geqslant 7$. Consequently, we obtain infinitely many solutions to the singular constant GJMS equation on round spheres $\mathbb{S}^n\setminus \mathbb{S}^k$ blowing up along a minimal equatorial subsphere with $0 \leqslant k<\frac{n-6}{2}$; this dimension restriction is sharp in the topological sense.