A note on scalar curvature type problems in dimension 4
type: Item
instance of: scholarly article
title: A note on scalar curvature type problems in dimension 4
zbMATH Open document ID: 1147.53033
author: Ezequiel R. Barbosa
author: Marcos Montenegro
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-06-17T00:00:00Z
review text: For a 4-dimensional compact Riemannian manifold \((M,g)\) let \(a\), \(f\) be positive \(C^\infty\)-functions. It is proved that the problem \(\Delta_g u+ a(x)u= f(x)u^3\) always admits a positive solution, up to a conformal deformation of \(g\). This leads to a geometric obstruction result for the prescribed scalar curvature \(S_g\) problem, which can be stated as follows: Let \(f\) be a nonzero, nonnegative function and let \(a={1\over 6} S_g\). If the above problem admits a positive solution for some \(h\) isometric to \(g\), then there exists no metric \(g_0\in [g]\) such that \(S_{g_0}\leq 0\) on \(M\).
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5289177
zbMATH Keywords: prescribed scalar curvature problem
zbMATH Keywords: geometric obstruction
zbMATH Keywords: curvature restrictions
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.01.106
OpenAlex ID: W2083706102
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DOI: 10.1016/J.JMAA.2008.01.106
description: scientific article; zbMATH DE number 5289177