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arXiv · 2505.23440

Comparison of total $\sigma_k$-curvature

Jiaqi Chen·Yufei Shan·Yinghui Ye
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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.

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Jiaqi Chen, Yufei Shan, Yinghui Ye. 2025-05-29. Comparison of total $\sigma_k$-curvature. https://arxiv.org/abs/2505.23440

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