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

Mingxiang Li, Juncheng Wei, Xingwang Xu

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 $σ_k(g)$ 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.

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

Abstract

The main purpose of current article is to study the geometry of -curvature. For simplicity, we start with a simple model: a complete and conformal metric on . Assuming that the metric has non-negative -order -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 elementary symmetric function of Ricci curvature over geodesic ball of radius is at most polynomial in with order for all . Similarly, we are able to show that the same growth control holds for -order -curvature. Finally, we show that for or , the gap theorems for hold true.

Paper Structure

This paper contains 8 sections, 43 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Logarithmic potential estimates and normal metrics
  3. Conformal metrics with non-negative scalar curvature
  4. Positivity of $Q$-curvatures and Ricci curvature
  5. Comparison of metric distance and measure distance
  6. Average growth rate of $\sigma_k$ and $Q^{(2k)}_g$
  7. Gap theorems for $Q^{(2k)}_g$-curvature
  8. Some remarks on the vanishing isoperimetric ratio

Key Result

Theorem 1.1

Let $g=e^{2u}|dx|^2$ be a complete and conformal metric on $\mathbb{R}^n$ with $n\geq 4$ an even integer. Suppose that its scalar curvature $R_g$ is non-negative outside a compact set and the negative part of $nth$-order $Q^{(n)}_g$-curvature is integrable over $(\mathbb{R}^n,g)$. Then, the total $Q where $|\mathbb{S}^n|$ denotes the volume of standard n-sphere and $\mathrm{d}\mu_g=e^{nu}\mathrm{d

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 63 more