A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866918390566748160 |
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| author | Ge, Yuxin Wang, Guofang Wei, Wei |
| author_facet | Ge, Yuxin Wang, Guofang Wei, Wei |
| contents | In this paper we introduce the following Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$: Find a conformal metric $g$ in a given conformal class $[g_0]$ with
\[
Q_g/R_g=const.
\] When the dimension $n\ge 5$, we first prove a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature on $\mathbb{S}^n$ ($n\ge 5$), namely
\[\frac{\int_{\mathbb{S}^n} Q_g d v_g}{\left(\int_{\mathbb{S}^n} R_g d v_g\right)^{\frac{n-4}{n-2}}} \geq \frac{\int_{\mathbb{S}^n} Q_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)}{\left(\int_{\mathbb{S}^n} R_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)\right)^{\frac{n-4}{n-2}}}\] for any $g$ in the conformal class of the round metric $g_{\mathbb{S}^n}$ with positive scalar curvature, with equality if and only if $g$ is also a metric with constant sectional curvature. With this inequality we introduce a new Yamabe constant $Y_{4,2}(M,[g_0])$ and prove the existence of the above problem provided that $Y_{4,2}(M,[g_0]) <Y_{4,2} (\mathbb{S}^n, [g_{\mathbb{S}^n}]).$ This strict inequality is proved if $(M,g)$ is not conformally equivalent to the round sphere. This follows from a crucial relation between $Y_{4,2}$ and the ordinary Yamabe constant $Y(M,[g_0])$, $Y_{4,2} (M, [g_0]) \le c(n) Y(M, [g_0])^{\frac n{n-2}}$ with equality if and only if $(M, g_0)$ is conformally equivalent to an Einstein manifold. Finally, we prove that on a closed $n$-dimensional Riemannian manifold $(M,g_{0})$ with semi-positive $Q$-curvature and non-negative scalar curvature, the above Yamabe problem is solvable, thanks to the maximum principle of Gursky-Malchiodi [33]. The proof for $n=3$ and $n=4$ follows closely the methods developed by Hang-Yang in [40], Gursky-Malchiodi in [33], and Chang-Yang in [12]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15074 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature Ge, Yuxin Wang, Guofang Wei, Wei Differential Geometry 53C21, 53C18, 58J05 In this paper we introduce the following Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$: Find a conformal metric $g$ in a given conformal class $[g_0]$ with \[ Q_g/R_g=const. \] When the dimension $n\ge 5$, we first prove a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature on $\mathbb{S}^n$ ($n\ge 5$), namely \[\frac{\int_{\mathbb{S}^n} Q_g d v_g}{\left(\int_{\mathbb{S}^n} R_g d v_g\right)^{\frac{n-4}{n-2}}} \geq \frac{\int_{\mathbb{S}^n} Q_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)}{\left(\int_{\mathbb{S}^n} R_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)\right)^{\frac{n-4}{n-2}}}\] for any $g$ in the conformal class of the round metric $g_{\mathbb{S}^n}$ with positive scalar curvature, with equality if and only if $g$ is also a metric with constant sectional curvature. With this inequality we introduce a new Yamabe constant $Y_{4,2}(M,[g_0])$ and prove the existence of the above problem provided that $Y_{4,2}(M,[g_0]) <Y_{4,2} (\mathbb{S}^n, [g_{\mathbb{S}^n}]).$ This strict inequality is proved if $(M,g)$ is not conformally equivalent to the round sphere. This follows from a crucial relation between $Y_{4,2}$ and the ordinary Yamabe constant $Y(M,[g_0])$, $Y_{4,2} (M, [g_0]) \le c(n) Y(M, [g_0])^{\frac n{n-2}}$ with equality if and only if $(M, g_0)$ is conformally equivalent to an Einstein manifold. Finally, we prove that on a closed $n$-dimensional Riemannian manifold $(M,g_{0})$ with semi-positive $Q$-curvature and non-negative scalar curvature, the above Yamabe problem is solvable, thanks to the maximum principle of Gursky-Malchiodi [33]. The proof for $n=3$ and $n=4$ follows closely the methods developed by Hang-Yang in [40], Gursky-Malchiodi in [33], and Chang-Yang in [12]. |
| title | A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature |
| topic | Differential Geometry 53C21, 53C18, 58J05 |
| url | https://arxiv.org/abs/2603.15074 |