Critical metrics for the L^2-norm of the curvature tensor

This dissertation explores the concept of $\mathcal{R}$-critical metrics, which are metrics that are critical points for the $L^2$-norm of the curvature tensor functional on a manifold. The functional is defined as the integral of the squared magnitude of the Riemann curvature tensor over the manifold. The research investigates the properties of these metrics, particularly in lower dimensions (three and four), and their relationship to Einstein metrics. The work presents a partial converse to the statement that Einstein metrics are $\mathcal{R}$-critical. Specifically, it shows that in four dimensions, if a metric has non-positive sectional curvature and is $\mathcal{R}$-critical, then it must be an Einstein metric. Furthermore, the dissertation provides classifications of $\mathcal{R}$-critical homogeneous spaces in dimensions three and four. This includes detailed analysis of left-invariant metrics on Lie groups, with specific results for unimodular groups and those with a non-trivial center.

Curriculum

The dissertation is structured to first introduce the motivation and key results, followed by a detailed review of differential geometry relevant to the study. The core of the work involves the classification of $\mathcal{R}$-critical homogeneous spaces in dimensions three and four.

1. 1993