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

TL;DR

This paper establishes a stability version of the almost-Schur lemma on the 3-sphere, leading to quantitative stability results for nonlinear Yamabe-type inequalities, including the $\sigma_2$-curvature functional, extending previous work to the critical case $d=3$.

Contribution

It proves a quantitative almost-Schur lemma on the 3-sphere and applies it to establish stability of Yamabe-type inequalities, including the $\sigma_2$-curvature, in the critical dimension.

Findings

Proved a stability inequality refining the Andrews-De Lellis-Topping inequality.

Established quantitative stability for a family of Yamabe-type inequalities in dimension 3.

Extended stability results to the critical case where the standard metric maximizes the $\sigma_2$-curvature functional.

Abstract

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.