On the variational properties of the prescribed Ricci curvature functional

TL;DR

This paper investigates the prescribed Ricci curvature problem on homogeneous spaces by analyzing the scalar curvature functional's properties, providing conditions for global maxima and illustrating results with specific examples.

Contribution

It introduces a variational approach to the prescribed Ricci curvature problem on homogeneous spaces, identifying conditions for the scalar curvature functional to attain global maxima.

Findings

Conditions for the scalar curvature functional to have a global maximum.

Analysis of the functional's behavior in specific homogeneous space examples.

Insights into solutions of the Ricci curvature equation for homogeneous metrics.

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.