A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature

Abstract

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].