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 (u): = \|g u - N-24 (N-1) Rg u + u^N+2{N-2}\|₇^-₁ (M) 0, then \|u- (u₀+₈=₁^ Vᵢ) \|₇℉ (₌) 0 where u₀ is a solution to the Yamabe problem on (M, g), 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+₈=₁^ Vᵢ) \|₇℉ (₌) C ( (u) ) where C > 0 and (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 (= 1) on generic Riemannian manifolds (M, g), proving that (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.