On symmetric scalar curvature on \(S^2\)
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.
scientific article; zbMATH DE number 1385318