Spectral theory of a conformal class
type: Item
instance of: scholarly article
title: Spectral theory of a conformal class
zbMATH Open document ID: 0842.53027
author: Yiren Li
published in: Bulletin des Sciences Mathématiques
publication date: 1996-07-28T00:00:00Z
review text: On a compact manifold the study of the conformal class \(C(g)\) of a Riemannian metric \(g\) consists of the determination of all metrics which possess a given property and of the determination of conformal invariants. In this paper the author studies the case of metrics with a given constant scalar curvature for a compact manifold \((V, g)\) of dimension \(n\geq 3\). Following the approaches of Yamabe, Aubin, Schoen, if \(C(g)\) is the set of the metrics \(g'\) with \(g'= \varphi^{4/(n- 2)} g\), the value of the scalar curvature \(R'\) of \(g'\) is a given constant \(\nu\) iff \(\varphi\) verifies the Yamabe equation \[ {4(n-1)\over n-2}\Delta\varphi+ R\varphi= \nu\varphi^{(n+ 2)/(n- 2)} \tag{1} \] where the solution can be normalized in such a way that \[ \int \varphi^p dV= 1,\tag{2} \] with \(p= 2n/(n-2)\). The spectrum \(\text{sp}(C)\) of the conformal class \(C\) is then defined as the set of all values \(\nu\) such that there exists a solution of (1) satisfying (2), and \(S_\nu\) indicates the set of all conformal metrics with scalar curvature \(\nu\) and volume 1. Moreover, if \(\mu(C)= \inf_{g'\in C} E(g')\) where \(E(g')\) is the functional energy of the conformal metric \(g'\), from results of Aubin, Yamabe, Schoen follows \(\mu(C)\in \text{sp}(C)\), and all the metrics of \(S_{\mu(C)}\) are called minimal Yamabe solutions. From the study of the Yamabe equation, the author proves that \(S(C)\) has the structure of an analytic set. He gives the cases in which the Yamabe problem has a unique solution and when a family of solutions corresponding to a family of metrics is relatively compact.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 851994
zbMATH Keywords: conformal invariants
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: Yamabe equation
zbMATH Keywords: Yamabe problem
reviewed by: Ida Cattaneo Gasparini
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 851994