The modified Yamabe problem and geometry of modified scalar curvatures
type: Item
instance of: scholarly article
title: The modified Yamabe problem and geometry of modified scalar curvatures
zbMATH Open document ID: 1078.53026
DOI: 10.1007/BF02921859
author: Mitsuhiro Itoh
published in: The Journal of Geometric Analysis
publication date: 2005-06-24T00:00:00Z
review text: In this paper, \((M,g)\) is a compact oriented \(n\)-manifold, \(n\geq 3\), equipped with a smooth metric \(g\) with scalar curvature \(s_g\) and Weyl conformal curvature \(W_g\). The modified Yamabe problem is a natural extension of the classical Yamabe problem proposed by \textit{M. Gursky} in his paper [Math. Ann. 318, No. 3, 417--431 (2000; Zbl 1034.53032)]. It is concerned with a modified scalar curvature which satisfies a conformal covariance condition. The problem is whether there exists a smooth metric \(\tilde g\) conformal to \(g\) such that this modified scalar curvature is constant. The modification adopted in this paper is a combination of the scalar curvature and the Weyl conformal curvature \(W_g\), depending on a certain real parameter \(t\), with \(t=0\) corresponding to the original Yamabe problem. The author proves an existence theorem analogous to a theorem of Trudinger and of Aubin [see \textit{N. Trudinger}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22, 265--274 (1968; Zbl 0159.23801) and \textit{T. Aubin}, J. Math. Pur. Appl., IX. Sér. 55, 269--296 (1976; Zbl 0336.53033)]. The author then obtains new results in dimension four by using modified curvature of a certain type and interpreting it geometrically. Thus, he obtains an extension of the four-dimensional sphere theorem of \textit{M. Micallef} and \textit{J. Moore} [Ann. Math. (2) 127, No. 1, 199--227 (1988; Zbl 0661.53027)]. Another modified scalar curvature is used in solving a problem concerning self-dual Einstein 4-manifolds, asking whether a compact negative Ricci, Einstein self-dual 4-manifold is either real or complex hyperbolic. The author also obtains results towards the Goldberg conjecture, which states that if a compact, almost Kähler \(2n\)-manifold is Einstein, then it must be Kähler. The result obtained, using a modified scalar curvature, is again under negative Ricci condition.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 2183023
zbMATH Keywords: modified scalar curvature
zbMATH Keywords: modified Yamabe problem
zbMATH Keywords: Weyl conformal curvature
MaRDI profile type: Publication
Wikidata QID: Q126281954
cites work: The Weyl functional near the Yamabe invariant
cites work: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
cites work: Q3657016
cites work: Q5749404
cites work: Q4720334
cites work: Entropy and rigidity of locally symmetric spaces of strictly negative curvature
cites work: Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein
cites work: Q3041921
cites work: The topology of four-dimensional manifolds
cites work: Four-manifolds with \(\delta W^+=0\) and Einstein constants of the sphere.
cites work: Four-manifolds with positive isotropic curvature
cites work: Compact four-dimensional Einstein manifolds
cites work: Kählerian Twistor Spaces
cites work: ALMOST KÄHLER 4-MANIFOLDS, L2-SCALAR CURVATURE FUNCTIONAL AND SEIBERG–WITTEN EQUATIONS
cites work: Uniformization of Kähler manifolds with vanishing Bochner tensor
cites work: Einstein metrics and Mostow rigidity
cites work: Q4400513
cites work: Kodaira dimension and the Yamabe problem
cites work: Ricci curvature, minimal volumes, and Seiberg-Witten theory
cites work: The Yamabe problem
cites work: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes
cites work: Metrics with nonnegative isotropic curvature
cites work: Conformal deformation of a Riemannian metric to constant scalar curvature
cites work: On some 4-dimensional compact Einstein almost Kähler manifolds
cites work: On some compact Einstein almost Kähler manifolds
cites work: A note on negative isotropic curvature
cites work: Curvature Tensors on Almost Hermitian Manifolds
cites work: Q5542608
Recommended article: On the Yamabe problem and the scalar curvature problems under boundary conditions
Recommended article: Q3135238
Recommended article: The Yamabe problem for higher order curvatures
Recommended article: Remarks on scalar curvature of Yamabe solitons
Recommended article: Q2960976
Recommended article: The Yamabe problem for distributional curvature
Recommended article: Yamabe metrics of positive scalar curvature and conformally flat manifolds
Recommended article: Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting
Recommended article: On scalar curvature of \(r\)-almost Yamabe solitons
Recommended article: Modified curvatures on manifolds with boundary and applications
description: scientific article; zbMATH DE number 2183023