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Comparison of total $σ_k$-curvature

Jiaqi Chen, Yufei Shan, Yinghui Ye

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 $σ_l$-curvature with respect to $σ_k$-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.

Comparison of total $σ_k$-curvature

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 -curvature with respect to -curvature (). In particular, we prove the comparison holds for metrics close to strictly stable positive Einstein metric with . As for negative Einstein metrics, we prove a similar comparison result provided certain assumptions on sectional curvature holds for the manifold.

Paper Structure

This paper contains 10 sections, 16 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Basic concepts
  4. Geometric variational formulas
  5. Basic variational formulas
  6. Variations for $\sigma_k$-curvature when $\bar{g}$ is Einstein
  7. The key functional and variations
  8. Total $\sigma_l$-curvature comparison with respect to $\sigma_k$-curvature
  9. Case 1: $k=1$
  10. Case 2: $k\geq2$

Key Result

Theorem 1.1

Suppose $(M^n,\bar{g})$ is a closed strictly stable Einstein manifold with where $\lambda\neq0$ is a constant. There exists a constant $\epsilon_0>0$ such that for any metric $g$ on $M$ which satisfies the following volume comparison holds: Moreover, the inequality holds in either case if and only if $g$ is isometric to $\bar{g}$.

Theorems & Definitions (31)

  • Theorem 1.1: Yuan, Yuan_VolumeCW
  • Theorem 1.2: Chen-Fang-He-Zhong2025Volume
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1: Einstein manifold
  • Definition 2.2: Stability of Einstein metric
  • Definition 2.3: $\sigma_k$-curvature
  • Lemma 2.1
  • Lemma 2.2
  • ...and 21 more