Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ0 with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem (P∈) —Lgu + ∈u = uN+2/N-2 in (M, g). We prove that for any k ∈ ℕ, there exists εk > 0 such that for all ε ∈ (0, εk) the problem (P∈) has a symmetric solution uε, which looks like the superposition of k positive bubbles centered at the point ξ0 as ε → 0. In particular, ξ0 is a towering blow-up point.