Equivariant Yamabe problem with boundary
type: Item
instance of: scholarly article
title: Equivariant Yamabe problem with boundary
zbMATH Open document ID: 1503.53064
author: Pak Tung Ho
author: Jinwoo Shin
published in: Calculus of Variations and Partial Differential Equations
publication date: 2022-02-16T00:00:00Z
review text: Let \((M,g)\) be a closed \(n\)-dimensional manifold with Riemannian metric \(g\) and \(n\ge 3\). The question of whether there is a metric on \(M\) conformal to \(g\) with constant scalar curvature is known as the Yamabe problem. \textit{E. Hebey} and \textit{M. Vaugon} [Bull. Sci. Math., II. Sér. 117, No. 2, 241--286 (1993; Zbl 0786.53024)] studied an equivariant extension of this problem. Namely, given a compact subgroup \(G\), of the isometry group of \(M\), does there exist a \(G\)-invariant metric conformal to \(g\) with constant scalar curvature? They were able to state conditions which imply an affirmative answer to this question. There are versions of the Yamabe problem that can be stated for manifolds with boundary as well as equivariant generalizations. Similar to the results of Hebey and Vaugon, for this equivariant generalization, the authors prove a sufficient condition that implies that the equivariant Yamabe problem for manifolds with boundary is solvable. As corollaries, they prove that this condition holds for certain manifolds with umbilic boundary which are locally conformally flat or have dimension \(3\le n\le 5\).
reviewed by: Joseph E. Borzellino
Mathematics Subject Classification ID: 53C18
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35R01
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 7474269
zbMATH Keywords: Yamabe problem
zbMATH Keywords: conformal structures
zbMATH Keywords: manifolds with boundary
Wikidata QID: Q114018026
MaRDI profile type: Publication
full work available at URL: s00526-021-02154-8
OpenAlex ID: W4206342259
cites work: A compactness theorem for scalar-flat metrics on manifolds with boundary
cites work: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
cites work: Q4398855
cites work: A generalization of the Yamabe flow for manifolds with boundary.
cites work: An existence theorem for the Yamabe problem on manifolds with boundary
cites work: Problèmes de Neumann non linéaires sur les variétés riemanniennes
cites work: Compactness and non-compactness for the Yamabe problem on manifolds with boundary
cites work: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary
cites work: The Yamabe problem on manifolds with boundary
cites work: Q4271944
cites work: The equivariant second Yamabe constant
cites work: On compact conformally euclidean spaces of dimension \(>2\)
cites work: Q4342727
cites work: Equivariant Yamabe problem and Hebey-Vaugon conjecture
cites work: Hebey-Vaugon conjecture. II
cites work: Barycenter technique and the Riemann mapping problem of Cherrier-Escobar
cites work: Conformal deformation of a Riemannian metric to constant scalar curvature
cites work: Convergence of the Yamabe flow for large energies
cites work: Surgery and equivariant Yamabe invariant
cites work: Q5542608
cites work: Conformal metrics on the unit ball with prescribed mean curvature
cites work: Q3275400
DOI: 10.1007/S00526-021-02154-8
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description: scientific article; zbMATH DE number 7474269