On geometry of $Q^{(2k)}_g$-curvature

TL;DR

This paper investigates the geometry of higher-order $Q$-curvature on conformal manifolds, establishing growth bounds and gap theorems under certain curvature and isoperimetric conditions.

Contribution

It provides new growth estimates for elementary symmetric functions of Ricci curvature and $Q$-curvature, and proves gap theorems for specific cases of $Q^{(2k)}_g$.

Findings

Ricci curvature is non-negative under given conditions.

Growth rate of symmetric functions of Ricci curvature is polynomial with degree $n-2k$.

Gap theorems hold for $Q^{(2k)}_g$ when $k=1$ or $2$.

Abstract

The main purpose of current article is to study the geometry of $Q$-curvature. For simplicity, we start with a simple model: a complete and conformal metric $g=e^{2u}|dx|^2$ on $\mathbb{R}^n$. Assuming that the metric $g$ has non-negative $nth$-order $Q$-curvature and non-negative scalar curvature, we show that the Ricci curvature is non-negative. If we further assume that the isoperimetric ratio near the end is positive, we show that the growth rate of $kth$ elementary symmetric function $\sigma_k(g)$ of Ricci curvature over geodesic ball of radius $r$ is at most polynomial in $r$ with order $n-2k$ for all $1 \leq k \leq \frac{n-2}{2}$. Similarly, we are able to show that the same growth control holds for $2kth$-order $Q$-curvature. Finally, we show that for $k=1$ or $2$, the gap theorems for $Q^{(2k)}_g$ hold true.