Multiple Blow-Up Phenomena for $Q$-Curvature in High Dimensions

TL;DR

This paper demonstrates the existence of infinitely many conformal metrics with constant $Q$-curvature and unbounded energy or volume on high-dimensional manifolds, extending previous scalar curvature results.

Contribution

It introduces a method to construct multiple blow-up solutions for the $Q$-curvature problem in high dimensions, generalizing earlier scalar curvature findings.

Findings

Existence of infinitely many metrics with constant $Q$-curvature and large energy.

Construction of sequences of metrics with constant $Q$-curvature and unbounded volume.

Extension of scalar curvature blow-up phenomena to the $Q$-curvature setting.

Abstract

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n \geq 25$ with positive Yamabe invariant $Y(M,g_0)>0$ and positive fourth-order invariant $Y_4(M,g_0)>0$. We show that, arbitrarily $C^1$-close to $g_0$, there exists a Riemannian metric such that, within its conformal class, one can find infinitely many smooth metrics with the same constant $Q$-curvature and arbitrarily large energy. Moreover, within this conformal class, there exists a sequence of smooth metrics with constant $Q$-curvature equal to $n(n^2-4)/8$ and unbounded volume. This extends to the $Q$-curvature setting the result previously obtained for the scalar curvature in Marques (2015) (see also Gond and Li (2025)). The proof is based on constructing small perturbations of multiple standard bubbles that are glued together.