Volume comparison theorem with respect to sigma-2 curvature

TL;DR

This paper establishes a volume comparison theorem related to sigma-2 curvature for metrics near stable Einstein metrics and proves local rigidity for Ricci-flat manifolds concerning sigma-2 curvature.

Contribution

It introduces a volume comparison theorem for sigma-2 curvature near stable Einstein metrics and proves local rigidity for Ricci-flat manifolds with respect to sigma-2 curvature.

Findings

Volume comparison theorem holds near stable Einstein metrics.

No positive sigma-2 curvature metrics near stable Ricci-flat metrics.

Rigidity results for Ricci-flat manifolds with respect to sigma-2 curvature.

Abstract

In this paper, we investigate the volume comparison theorem related to $\sigma_2$-curvature. In particular, we show that volume comparison theorem with respect to $\sigma_2$-curvature holds for metrics close to strictly stable positive Einstein metrics. By applying similar techniques, we derive the local rigidity theorem for strictly stable Ricci flat manifolds with respect to $\sigma_2$-curvature, which shows it admits no metric with positive $\sigma_2$-curvature near strictly stable Ricci-flat metrics.