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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
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