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.


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