The sharp $\sigma_2$-curvature inequality on the sphere in quantitative form

TL;DR

This paper proves a stability version of the sharp $\sigma_2$-curvature inequality on the sphere, showing that metrics nearly minimizing the normalized total $\sigma_2$-curvature are close to the standard metric in Sobolev norms.

Contribution

It establishes the optimal stability exponents for metrics nearly minimizing the $\sigma_2$-curvature inequality on spheres, extending the understanding of the inequality's rigidity.

Findings

Metrics close to minimizing the $\sigma_2$-curvature are close to the standard metric in Sobolev norms.

The stability exponents for the closeness are optimal.

The result applies to fully nonlinear Euler-Lagrange equations.

Abstract

Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $\sigma_2$-curvature, normalized by the volume, is uniquely (up to M\"obius transformations) minimized by the standard metric. We show that if a metric almost minimizes, then it is almost the standard metric (up to M\"obius transformations). This closeness is measured in terms of Sobolev norms of the conformal factor, and we obtain the optimal stability exponents for two different notions of closeness. This is a stability result for an optimization problem whose Euler-Lagrange equation is fully nonlinear.