The Yamabe problem on CR manifolds
type: Item
instance of: scholarly article
title: The Yamabe problem on CR manifolds
zbMATH Open document ID: 0661.32026
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: The Yamabe problem is to find a metric of constant scalar curvature in a given conformal class of Riemannian metrics. This paper studies the analogous problem for strictly pseudoconvex CR-manifolds. The notion of scalar curvature used here is due to S. Webster. After introducing the relevant variational problem for CR manifolds an invariant \(\lambda\) (N) is defined to be the infimum of the functional under consideration. This corresponds in the Riemannian case to the infimum of the total scalar curvature in a given conformal class with fixed volume. The main result of the paper proves that if \(N^{2n+1}\) is a compact, integrable strictly pseudoconvex CR manifold with \(\lambda (N)<\lambda (S^{2n+1}),\) then the infimum is attained by a solution of constant Webster scalar curvature. The proof is modelled closely on the solution of the Riemannian Yamabe problem in the ``easy'' case due to N. Trudinger and T. Aubin. The analytical estimates are a bit more involved in the CR case and are based on work of G. Folland and E. Stein.
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 4079881
zbMATH Keywords: Sobolev inequalities
zbMATH Keywords: Fefferman metric
zbMATH Keywords: subelliptic estimates
zbMATH Keywords: Heisenberg group
zbMATH Keywords: pseudoconvex CR-manifolds
zbMATH Keywords: Webster scalar curvature
zbMATH Keywords: Riemannian Yamabe problem
author: David Jerison
author: John M. Lee
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214440849
OpenAlex ID: W1509786127
Wikidata QID: Q115184837
DOI: 10.4310/JDG/1214440849
description: scientific article; zbMATH DE number 4079881