A sharp Sobolev inequality on Riemannian manifolds

Abstract

Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 6$. We prove that <p align="center"> $||u||_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_{g} u|^2+c(n)R_{g} u^2\}dv_g +A||u||_{L^{2n/(n+2)}(M,g)}^2,$ <p align="left" class="times"> for all $u\in H^1(M)$, where $2^*=2n/(n-2)$, $c(n)=(n-2)/[4(n-1)]$, $R_g$ is the scalar curvature, $K^{-1}=$ inf $\|\nabla u\|_{L^2(\mathbb R^n)}\|u\|_{L^{2n/(n-2)}(\mathbb R^n)}^{-1}$ and $A>0$ is a constant depending on $(M,g)$ only. The inequality is <em>sharp</em in the sense that on any $(M,g)$, $K$ can not be replaced by any smaller number and $R_g$ can not be replaced by any continuous function which is smaller than $R_g$ at some point. If $(M,g)$ is not locally conformally flat, the exponent $2n/(n+2)$ can not be replaced by any smaller number. If $(M,g)$ is locally conformally flat, a stronger inequality, with $2n/(n+2)$ replaced by $1$, holds in all dimensions $n\ge 3$.