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 $\mathrm{Ric}(g)=cT$ 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 for a general homogeneous space under which it has a global maximum. Finally, we study the behavior of the functional in specific examples to illustrate our result.

24 pages, 3 figures. Version 2: A portion of this paper has been generalised and moved to the new paper "Palais-Smale sequences for the prescribed Ricci curvature functional"; the current version deals with the question of when the prescribed Ricci curvature functional assumes its maximum

References in corpus (4)

• On the Ricci iteration for homogeneous metrics on spheres and projective spaces
• Invariant Einstein metrics on Ledger-Obata spaces
• The Prescribed Ricci Curvature Problem on Homogeneous Spaces with Intermediate Subgroups
• Diverging sequences of unit volume invariant metrics with bounded curvature