Conformal metrics of prescribed scalar curvature on 4-manifolds: the degree zero case
Given a compact, \(4\)-dimensional Riemannian manifold \((M^4,g_0)\) with positive scalar curvature \(R_{g_0},\) and a function \(K\in C^2,\) the prescribed scalar curvature problem consists of finding a metric \(g,\) conformally related to \(g_0,\) such that the scalar curvature of \((M, g)\) equals \(K.\) Setting \(g = u^2 g_0,\) this leads to the problem \[ L_{g_0}u=K u^3,\quad u>0\;\text{on}\;M^4, \] where \(L_{g_0}\) stands for the conformal Laplacian \(-\Delta_{g_0}+1/6.\) The authors derive existence results for the above problem that generalize in some sense these already known in the literature. The technique applied relies on estimates for the Morse index of solutions.
scientific article; zbMATH DE number 6822574
scientific article