Conformal deformation of a Riemannian metric to constant scalar curvature
type: Item
instance of: scholarly article
title: Conformal deformation of a Riemannian metric to constant scalar curvature
zbMATH Open document ID: 0576.53028
published in: Journal of Differential Geometry
publication date: 1984-01-01T00:00:00Z
review text: The author announces the existence of metrics of constant scalar curvature in any conformal class on compact manifolds which are conformally flat near a point or which are 3-, 4- or 5-dimensional. This together with previous arguments due to Thierry Aubin settles the proof of the Yamabe conjecture. Two new ideas are involved. Assuming first that the metric is conformally flat near a point, the author constructs a function on which the Yamabe functional takes on a value below the critical one. This function solves an exterior boundary value problem near that point. Its existence relies on an extension to any dimension of the theorem due to S. T. Yau and the author asserting that the total mass is positive for asymptotically Euclidean metrics with nonnegative scalar curvature. (This part of the proof is not yet in print.) Secondly, a delicate perturbation argument working in dimensions 3, 4 and 5 allows the author to get rid of the assumption of conformal flatness at a point.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 3921414
zbMATH Keywords: conformal class
zbMATH Keywords: conformally flat near a point
zbMATH Keywords: Yamabe conjecture
zbMATH Keywords: total mass
author: Richard Schoen
reviewed by: Jean Pierre Bourguignon
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214439291
OpenAlex ID: W2159531493
Wikidata QID: Q115187454
DOI: 10.4310/JDG/1214439291
description: scientific article; zbMATH DE number 3921414