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 ξ_0 with non-vanishing Weyl’s tensor. We consider a linear perturbation of the Yamabe problem in (M, g). We prove that for any k ∈ ℕ, there exists ε_k > 0 such that for all ε ∈ (0, ε_k) the problem has a symmetric solution u_ε, which looks like the superposition of k positive bubbles centered at the point ξ_0 as ε → 0. In particular, ξ_0 is a towering blow-up point.