Comparison of total quotient curvature
Jiaqi Chen, Yi Fang, Jingyang Zhong
TL;DR
The paper addresses the problem of comparing total quotient curvature functionals near a strictly stable Einstein metric. It develops a variational framework for the quotient curvatures $\frac{\sigma_k}{\sigma_l}$ by defining a scale-invariant key functional $\mathcal{H}_{\bar{g}}(g)$ and deriving its first- and second-order variations, establishing that $\bar{g}$ is a critical point and that the second variation is non-positive under explicit index conditions. Using a slice through the background metric, Morse-type rigidity, and spectral estimates (Bochner and Lichnerowicz-Obata), it proves sharp integral inequalities: if $\frac{\sigma_k}{\sigma_l}(g)$ bounds hold in the appropriate direction, then $\int_M \frac{\sigma_p(g)}{\sigma_q(g)}\, dv_g \le (\text{or} \ge) \int_M \frac{\sigma_p(\bar{g})}{\sigma_q(\bar{g})}\, dv_{\bar{g}}$, with equality forcing isometry to $\bar{g}$. This extends scalar-volume and $\sigma_k$-curvature volume comparisons to total quotient curvature and tightens rigidity statements for metrics near strictly stable Einstein backgrounds, offering a nonlinear, conformally informed analogue of classical volume comparison results.
Abstract
In this paper, we establish some comparison theorems for the total quotient curvature. Specifically, we examine the behavior of the functional with respect to the total quotient curvature and prove that the background Einstein metric achieves a sharp bound on the total quotient curvature. We prove that if the quotient curvature satisfies a point-wise lower (or upper) bound relative to the Einstein metric, then the corresponding integral inequality holds. Also we can show characterize the equality case. Our result generalizes the volume comparison theorem for scalar curvature and the rigidity results for $σ_k$-curvature.