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We know Schoen's compactness on Yamabe problem:

Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension $3\leq n\leq 24$ without boundary. Denote $\Phi$ to be the full set of arbitrary solutions for the Yamabe equation: $\Phi=\{\varphi>0|4\frac{n-1}{n-2}\Delta\varphi+S\varphi=Q_g(\varphi)\varphi^{\frac{n+2}{n-2}},\|\varphi\|_{\frac{2n}{n-2}}=1\}$.

If $(M^n,g)$ has positive Yamabe quotient and is not conformally diffeomorphic to $(S^n,g_{std})$,then there exists a constant $C>0$ depending only on $g$ such that $\frac{1}{C}\leq \varphi\leq C\quad \hbox{and} \quad\|\varphi\|_{C^{2,\alpha}}\leq C,\quad \forall \varphi \in \Phi.$

My question is that I want to estimate the constant $C$ more precisely, or at least for the Yamabe minimizer? In other words, under what assumption on the metric (such as Yamabe constant, Scalar curvature bound, volume, etc), we have an estimate on the Yamabe minimizer?

Further more, In the nonpositive Yamabe constant case, can we estimate the unique solution for the Yamabe equation with the information of the Scalar curvature bound (Not necessary nonpositive), Yamabe constant, etc?

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You mean the compactness result of Khuri, Marques and Schoen? –  YangMills Apr 14 '12 at 3:16
The Khuri-Marques-Schoen proof is by blowup arguments. It is general fact of life that arguments by blowup/contradiction don't give very explicit constants... –  Robert Haslhofer Apr 14 '12 at 7:53
For the last question, you might benefit from Green's function estimates for Schrödinger type equations. Please let me know if you find something. –  timur Apr 24 '12 at 4:32
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