On geometry of $Q^{(2k)}_g$-curvature
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917068432998400 |
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| author | Li, Mingxiang Wei, Juncheng Xu, Xingwang |
| author_facet | Li, Mingxiang Wei, Juncheng Xu, Xingwang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20165 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On geometry of $Q^{(2k)}_g$-curvature Li, Mingxiang Wei, Juncheng Xu, Xingwang Differential Geometry Analysis of PDEs 53C18, 58J90 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. |
| title | On geometry of $Q^{(2k)}_g$-curvature |
| topic | Differential Geometry Analysis of PDEs 53C18, 58J90 |
| url | https://arxiv.org/abs/2506.20165 |