Remarque sur le problème de Yamabe. (Remark on the problem of Yamabe)
Let (X,g) be a compact Riemannian manifold of dimension \(n\geq 3\), with scalar curvature R. Let I(\(\psi\)) be the functional on the Sobolev space \(H^ 2_ 1(X)\) defined by \[ I(\psi)=(\int \psi^ Ndv)^{-2/N}(\int | \nabla \psi |^ 2dv+[(n-2)/4(n-1)]\int R\psi^ 2dv),\text{ where } N^ 2=2n/(n-2). \] It has been proved by T. Aubin that if the infimum of I(\(\psi\)) over all positive non identically zero functions of \(H^ 2_ 1(X)\) is strictly less than \(n(n-2)\omega_ n^{2/n}/4,\) (where \(\omega_ n\) is the area of the unit sphere \(S^ n\) in \({\mathbb{R}}^{n+1})\), then one can find a metric \(g'\) with scalar curvature 1, conformal to g. The authors construct suitable functions which show that I(\(\psi\)) satisfies this strict inequality.
scientific article; zbMATH DE number 4215005
scientific article