Comparison of total quotient curvature

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.

💡 Research Summary

The paper investigates comparison theorems for a fully nonlinear curvature invariant called the total quotient curvature, defined as the ratio σₖ/σ_ℓ of elementary symmetric functions of the Schouten tensor’s eigenvalues. Working on a closed n‑dimensional Riemannian manifold (Mⁿ,g), the authors focus on a background Einstein metric (\bar g) satisfying (\operatorname{Ric}{\bar g}=(n-1)\lambda\bar g) with (\lambda>0). They require that (\bar g) be strictly stable, meaning that the Einstein operator (\Delta{\bar g}^E=\Delta_{\bar g}+2\operatorname{Rm}_{\bar g}) is strictly negative on the space of transverse‑traceless symmetric 2‑tensors. This stability condition guarantees a definite sign for the second variation of certain curvature functionals.

The central result (Main Theorem A) states that for integers (1\le\ell<k\le n) and (1\le q\le p\le n) there exists a small constant (\varepsilon_0>0) such that any metric (g) with (|g-\bar g|{C^2}<\varepsilon_0) satisfies a global integral inequality for the total quotient curvature (\int_M\sigma_p(g)\sigma_q(g),dv_g) provided a pointwise bound on the lower‑order quotient (\sigma_k(g)\sigma\ell(g)) holds. Precisely, if either

1. (\sigma_k(g)\sigma_\ell(g)\ge\sigma_k(\bar g)\sigma_\ell(\bar g)) and (2(p-q)<n), or
2. (\sigma_k(g)\sigma_\ell(g)\le\sigma_k(\bar g)\sigma_\ell(\bar g)) and (2(p-q)\ge n+2(k-\ell)),

then
\