## A priori estimates for the Yamabe problem in the non-locally conformally flat case.(English)Zbl 1101.53019

The author considers the set of solutions to the Yamabe problem. Given a compact Riemannian manifold $$(M^n,g),$$ $$n\geq 3,$$ without boundary, find a metric $$\widetilde{g}$$ of constant scalar curvature conformally related to $$g.$$ This is equivalent to the existence of a positive solution $$u$$ to the equation $$\Delta _g u-\frac{n-2}{4(n-1)} R_g u+Ku^\frac{n+2}{n-2} =0$$ on $$M,$$ where $$\Delta_g u$$ is the Laplace-Beltrami operator for the metric $$g$$, $$R_g$$ is the scalar curvature on $$M$$ and $$K$$ is a constant.
The main result of the paper is to find a priori estimates for the solutions to the Yamabe equation on Riemannian manifolds of low dimension. In particular, if $$( M^{n},g),$$ $$n\leq 7,$$ is a smooth closed Riemannian manifold with positive Yamabe quotient and not conformally equivalent to $$(\mathbb{S}^n,g_0),$$ then given $$\varepsilon >0$$ there exists a positive constant $$C=C(\varepsilon ,g)$$ such that $$(1/C)\leq u\leq C$$ and $$\| u\| _{C^{2,a}(M)}\leq C$$, $$0<a<1,$$ for every $$u\in \bigcup _{1+\varepsilon \leq p\leq \frac{n+2}{n-2}} \mathbf{M}_{p}$$, where $$\mathbf{M}_p=\{ u>0\mid \Delta _g u- \frac{n-2}{4(n-1)}R_g u+Ku^p=0\}$$. The proof uses the positive mass theorem which holds in dimension $$n\leq 7$$. As a consequence, the set of solutions of the Yamabe equation in the $$C^{2}$$-topology is compact. It is also shown that in dimension $$n\geq 6$$ the Weyl tensor vanishes at points where solutions to the Yamabe equation blow up.

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

### Keywords:

Yamabe problem; a priori estimates; positive mass theorem
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