On critical Riemannian metrics for a curvature functional on 3-manifolds
type: Item
instance of: scholarly article
title: On critical Riemannian metrics for a curvature functional on 3-manifolds
zbMATH Open document ID: 1017.58009
author: Minyo Katagiri
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-08-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1148392730
review text: One approach to geometrization problems in 3-manifolds \(M\) is to consider all Riemannian metrics \({\mathcal M}(M)\) on \(M\), to define a functional \(F: {\mathcal M}(M)\to\mathbb R\), and to pick out nice metrics on \(M\) as critical points of the functional. There are difficulties in the procedure: (1) Existence Theorems. Do there exist critical points? (2) Adequacy of the Functional. Does the functional control convergence or divergence of metrics? Of curvature of the metrics? Other properties of the metric? The author considers as functional \(F\) a normalized \(L^2\)-norm of the traceless part of the Ricci curvature. He notes that Einstein metrics are critical points for his functional \(F\). In regard to existence of critical points for the functional, the author constructs a critical metric for \(F\) on the 3-sphere \(\mathbb S^3\) which is not Einstein. He proves that, if a metric is critical for \(F\) and satisfies an additional positivity condition, then the metric is Einstein.
Mathematics Subject Classification ID: 58E11
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1775620
zbMATH Keywords: Riemannian metric
zbMATH Keywords: curvature
zbMATH Keywords: critical metrics
zbMATH Keywords: Einstein metrics
reviewed by: James W. Cannon
MaRDI profile type: Publication
OpenAlex ID: W2031496059
Wikidata QID: Q115219915
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DOI: 10.3792/PJAA.78.43
description: scientific article; zbMATH DE number 1775620