# Optimal Sobolev Inequality

Recall the Optimal Sobolev Inequality: Let $(M^n,g)$ be a smooth, compact Riemannian $n (\geq 3)$ manifold with $\hbox{inj}_g\geq i_0, |Ric(g)|\leq \Lambda g$. Let $\sigma_n=\frac{4(n-1)}{n-2}\frac{1}{\Lambda(S^n)}$ be the sharp Sobolev constant of $\mathbb{R}^n$, $\gamma=\frac{2n}{n-2}$. There exists $C=C(n,\Lambda, i_0)$ such that for any $\varphi \in W^{1,2}(M)$, $(\int_M|\varphi|^{2\gamma})^{\frac{1}{\gamma}}\leq\sigma^2_n\int_M|\nabla\varphi|^2+C\int_M\varphi^2.$

In particular, If $(M,g)$ admits a conformal immersion into $(S^n,g_{std})$ ($\Leftarrow$ simply connected and conformal flat), then $(\int_M|\varphi|^{2\gamma})^{\frac{1}{\gamma}}\leq \sigma^2_n\int_M|\nabla\varphi|^2+\frac{n-2}{4(n-1)}||S||_{L^{\infty}}\int_M\varphi^2.$

Question: Can we estimate $C$ in terms of $C_S$ and $||S||_{L^{\infty}}$, where $S$ is the Scalar curvature and $C_S$ is usual Sobolev constant: $(\int_M|\varphi|^{2\gamma})^{\frac{1}{\gamma}}\leq C_S(\int_M|\nabla\varphi|^2+\varphi^2)$?

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