An almost-almost-Schur lemma on the 3-sphere

In the conformal class of the standard metric on the $3$-sphere, we prove a quantitative refinement of the Andrews-De Lellis-Topping inequality in terms of a two-term distance to the set of minimizing conformal factors. This inequality is itself a stability result for the well-known Schur lemma and is therefore referred to as almost-Schur lemma. Hence, our stability result may be viewed as an almost-almost-Schur lemma. As a consequence, we deduce via interpolation the quantitative stability of an entire family of nonlinear Yamabe-type inequalities, including an inequality for the total volume-normalized $\sigma_2$-curvature $\mathcal F_2$. This extends a recent result by Frank and the second author for $d > 4$ to the case $d=3$. While the standard metric minimizes $\mathcal F_2$ if $d > 4$, it maximizes $\mathcal F_2$ if $d=3$. This is the main challenge in treating the case $d=3$ as it turns the related functional inequality into a reverse Sobolev-type inequality.