Sharp quantitative stability of the Yamabe problem

TL;DR

This paper establishes sharp quantitative stability estimates for the Yamabe problem on closed Riemannian manifolds, revealing differences in behavior between low and high dimensions and extending prior results to more general settings.

Contribution

The authors construct optimal stability estimates for the Yamabe problem on general manifolds, including new results for dimensions 3-5 and insights into single-bubbling phenomena in higher dimensions.

Findings

Stability estimates are sharp and dimension-dependent.

For 3 ≤ N ≤ 5, constructed suitable bubble functions and proved linear stability.

For N ≥ 6, analyzed single-bubble cases and showed stability can be much larger than linear.

Abstract

Given a smooth closed Riemannian manifold $(M,g)$ of dimension $N \ge 3$, 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) \cite{S} states that if $\Gamma(u) := \|\Delta_g u - \frac{N-2}{4(N-1)} R_g u + u^{\frac{N+2}{N-2}}\|_{H^{-1}(M)} \to 0$, then $\|u-(u_0+\sum_{i=1}^{\nu} \mathcal{V}_i)\|_{H^1(M)} \to 0$ where $u_0$ is a solution to the Yamabe problem on $(M,g)$, $\nu \in \mathbb{N} \cup \{0\}$, and $\mathcal{V}_i$ is a bubble-like function. If $M$ is the round sphere $\mathbb{S}^N$, then $u_0 \equiv 0$ and a natural candidate of $\mathcal{V}_i$ is a bubble itself. If $M$ is not conformally equivalent to $\mathbb{S}^N$, then either $u_0 > 0$ or $u_0 \equiv 0$, there is no canonical choice of $\mathcal{V}_i$, and so a careful selection of $\mathcal{V}_i$ must be made to attain optimal estimates. For $3 \le N \le 5$, we construct suitable $\mathcal{V}_i$'s and then establish the inequality $\|u-(u_0+\sum_{i=1}^{\nu} \mathcal{V}_i)\|_{H^1(M)}$ $ \le C\zeta(\Gamma(u))$ where $C > 0$ and $\zeta(t) = t$, consistent with the result of Figalli and Glaudo (2020) \cite{FG} on $\mathbb{S}^N$. In the case of $N \ge 6$, we investigate the single-bubbling phenomenon $(\nu = 1)$ on generic Riemannian manifolds $(M,g)$, proving that $\zeta(t)$ is determined by $N$, $u_0$, and $g$, and can be much larger than $t$. This exhibits a striking difference from the result of Ciraolo, Figalli, and Maggi (2018) \cite{CFM} on $\mathbb{S}^N$. All of the estimates presented herein are optimal.