Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere

In this work, we focus on a recent variant of the Trudinger-Moser-Onofri inequality introduced by S. Y. Alice Chang and Changfeng Gui \cite{CG-2023}: \begin{align*} α\int_{$\mathbb{S}$^2}|\nabla_{$\mathbb{S}$^2}u|^2 {\rm d}ω+2 \int_{$\mathbb{S}$^2} u {\rm d}ω-\frac{1}{2}\ln\left[\left(\int_{$\mathbb{s}$^2}e^{2u}{\rm d}ω\right)^2-\sum_{i=1}^3\left(\int_{$\mathbb{s}$^2}ω_i e^{2u}{\rm d} ω\right)^2\right] \geq 0 \end{align*} holds on $H^1({\mathbb{S}}^2)$ if and only if $\alpha \ge \frac{2}{3}$. In this regime, the infimum is attained only by trivial functions when $\alpha >\frac{2}{3},$ whereas for the critical value $\alpha =\frac{2}{3}$ nontrivial extremals exist, and Chang-Gui further provided a complete classification of such solutions. Building upon their result, we found a nice conformal invariance of the associated functional. Exploiting this invariance, we were able to characterize the full family of extremals in terms of conformal maps of ${\mathbb{S}}^2$ and, moreover, establish a sharp quantitative stability result in the gradient norm.