Sharp quantitative stability of the Yamabe problem

Given a smooth closed Riemannian manifold $(M,g)$ of dimension N ≥ 3, we derive sharp quantitative stability estimates for nonnegative functions near the solution set of the Yamabe problem on $(M,g)$. The seminal work of Struwe (1984) {S} states that if Γ(u) := \|Δg u - N-2/4(N-1) Rg u + uN+2/N-2\|_H⁻¹(M) → 0, then \|u-(u₀+∑ᵢ₌₁ν Vᵢ)\|H¹(M) → 0 where u₀ is a solution to the Yamabe problem on $(M,g)$, ν ∈ N ∪ \0\, and Vᵢ is a bubble-like function. If M is the round sphere SN, then u₀ ≡ 0 and a natural candidate of Vᵢ is a bubble itself. If M is not conformally equivalent to SN, then either u₀ > 0 or u₀ ≡ 0, there is no canonical choice of Vᵢ, and so a careful selection of Vᵢ must be made to attain optimal estimates. For 3 ≤ N ≤ 5, we construct suitable Vᵢ's and then establish the inequality \|u-(u₀+∑ᵢ₌₁ν Vᵢ)\|H¹(M) ≤ Cζ(Γ(u)) where $C > 0$ and ζ(t) = t, consistent with the result of Figalli and Glaudo (2020) {FG} on SN. In the case of N ≥ 6, we investigate the single-bubbling phenomenon (ν = 1) on generic Riemannian manifolds $(M,g)$, proving that ζ(t) is determined by N, u₀, and g, and can be much larger than t. This exhibits a striking difference from the result of Ciraolo, Figalli, and Maggi (2018) {CFM} on SN. All of the estimates presented herein are optimal.