On the variational properties of the prescribed Ricci curvature functional

Abstract

We study the prescribed Ricci curvature problem for homogeneous metrics. Given a (0,2)-tensor field T , this problem asks for solutions to the equation Ric(g) = cT for some constant c. Our approach is based on examining global properties of the scalar curvature functional whose critical points are solutions to this equation. We produce conditions under which it has a global maximum and find a large of class of spaces where it admits saddle critical points, even though the Palais-Smale condition fails to hold. Finally, we consider several examples, including ones where the functional admits critical submanifolds of various dimensions.