Remarque sur le problème de Yamabe. (Remark on the problem of Yamabe)
type: Item
instance of: scholarly article
title: Remarque sur le problème de Yamabe. (Remark on the problem of Yamabe)
zbMATH Open document ID: 0734.58044
DOI: 10.1016/0022-1236(91)90071-C
author: Emmanuel Hebey
author: Michel Vaugon
published in: Journal of Functional Analysis
publication date: 1991-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4215005
zbMATH Keywords: Yamabe problem
zbMATH Keywords: conformal metrics
reviewed by: Paul Godin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4215005