Comparison of total $\sigma_k$-curvature

TL;DR

This paper extends volume comparison theorems in Riemannian geometry to compare total $\sigma_l$-curvature and $\sigma_k$-curvature, especially near stable Einstein metrics, with results for both positive and negative Einstein metrics.

Contribution

It introduces new comparison results for total $\sigma_l$-curvature relative to $\sigma_k$-curvature, generalizing classical volume comparison theorems in Riemannian geometry.

Findings

Comparison holds near stable positive Einstein metrics for $l<rac{n}{2}$.

Similar comparison results for negative Einstein metrics under curvature assumptions.

Extension of classical volume comparison to $\sigma_l$-curvature contexts.

Abstract

Volume comparison theorem is a type of fundamental results in Riemannian geometry. In this article, we extend the volume comparison result in \cite{Besse2008} to the comparison of total $\sigma_l$-curvature with respect to $\sigma_k$-curvature ($l<k$). In particular, we prove the comparison holds for metrics close to strictly stable positive Einstein metric with $l<\frac{n}{2}$. As for negative Einstein metrics, we prove a similar comparison result provided certain assumptions on sectional curvature holds for the manifold.