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

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.

💡 Research Summary

The paper investigates the constant $Q$‑curvature problem on closed Riemannian manifolds of dimension $n\ge25$ under the positivity assumptions $Y(M,g_0)>0$ and $Y_4(M,g_0)>0$. These hypotheses guarantee that the Paneitz operator is positive definite, which provides a variational framework analogous to the classical Yamabe problem but of fourth order. The authors prove that for any $\varepsilon>0$ one can find a smooth metric $g$ with $|g-g_0|_{C^1}<\varepsilon$ such that within its conformal class $