On symmetric scalar curvature on \(S^2\)
type: Item
instance of: scholarly article
title: On symmetric scalar curvature on \(S^2\)
zbMATH Open document ID: 0935.53020
DOI: 10.1142/S0252959999000369
author: Min Ji
published in: Chinese Annals of Mathematics. Series B
publication date: 2000-05-03T00:00:00Z
review text: The scalar curvature problem on the unit sphere \(S^2\) is to determine conditions on a function \(R\) so that \(R\) corresponds to the scalar curvature function of a conformal metric \(e ^u g_0\), where \(g_0\) is the standard metric on \(S^2\) and \(u\) is a function on \(S^2\). The problem is equivalent to the solvability of the differential equation \((*)\) \(-\Delta u +2-R e ^u = 0\) on \(S^2\), where \(\Delta\) is the Laplacian operator on \(S^2\) with respect to \(g_0\). Let \(G\) be a subgroup of the orthogonal group of \(S^2\), and \(f_G\) be the set of fixed points of \(G\) on \(S^2\). Some new results are given by seeking a solution of minimax type, in the case where \(R\) is \(G\)-symmetric. Suppose, for example, that \(R\) is \(G\)-symmetric, \(R \in C^2 (S^2)\) and \(m = {\text{min}}R\) on \(R_G\) is positive. If there exists a point \(x_0 \in \{x \in f_G : R(x) = m \}\) such that \(\Delta R(x_0) <0\), then \((*)\) is solvable.
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 1385318
zbMATH Keywords: scalar curvature
zbMATH Keywords: symmetry
zbMATH Keywords: minimax method
reviewed by: Thomas Hasanis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1385318