The sharp ${\sigma }_k$-curvature inequality on locally conformally flat manifolds in quantitative form

Let $2\le k<n\mathrm{/}2$ and let $(M^n,[g_0])$ be a closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g_0]$. We prove a stability result of the ${\sigma }_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conformal metric, then this metric is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the $W^{1,2}$- and the $W^{1,2k}$-norm with optimal exponents $2$ and $2k$, respectively. This extends a previous result by Frank and the author to $2<k<n\mathrm{/}2$ and, under an additional non-degeneracy assumption, to the full class of manifolds originally considered by Viaclovsky.