Existence and compactness of minimizers of the Yamabe problem on manifolds with boundary
type: Item
instance of: scholarly article
title: Existence and compactness of minimizers of the Yamabe problem on manifolds with boundary
zbMATH Open document ID: 1070.53016
author: Henrique Araújo
published in: Communications in Analysis and Geometry
publication date: 2004-10-22T00:00:00Z
review text: Let \((M,g)\) be a Riemannian manifold with boundary. In [J. Differ. Geom. 35, No. 1, 21--84 (1992; Zbl 0771.53017)], \textit{P. Escobar} showed that, if the dimension of \(M\) is greater than or equal to three, \(M\) can be endowed with a metric \(\tilde g\) conformal to \(g\) having constant scalar curvature and minimal boundary. The total scalar curvature plus total mean curvature functional \(E\), defined on the space of Riemannian metrics of a compact manifold with boundary, arises in connection with this generalization of the Yamabe problem for manifolds with boundary. The critical points of \(E\) on a given conformal class of metrics (subject to various volume and area constraints) are metrics of constant scalar curvature and constant mean curvature on the boundary. In the present paper, the author considers the set of all metrics conformal to \(g\) and satisfying \(aV + bA =1\), where \(a\) and \(b\) are positive real numbers, \(V\) and \(A\) are the volume of \(M\) and area of the boundary of \(M\) respectively. He proves the existence of minimizers of the functional \(E\) restricted to the above set and when the infimum of \(E\) on that set is stricly less than the corresponding quantity on the standard Euclidean half-sphere. This allows to deform \(g\) conformally to a metric having constant scalar curvature \(R\) and constant mean curvature \(H\) on the boundary and such that \(R\) and \(H\) are subject to the relation \(bR =2nHa\). Moreover, the author establishes compactness results for the set of all minimizers when the base metric is allowed to vary on a small \(C^3\) neighbourhood in the set of Riemannian metrics.
reviewed by: Sergio Console
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58E11
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 58H15
zbMATH DE Number: 2109260
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Yamabe problem
zbMATH Keywords: scalar
MaRDI profile type: Publication
full work available at URL: cag.2004.v12.n3.a1
OpenAlex ID: W2004967502
DOI: 10.4310/CAG.2004.V12.N3.A1
description: scientific article; zbMATH DE number 2109260