A critical metric for the $L^2$-norm of the curvature tensor on $S^3$

Abstract:

The $L^2$-norm of the curvature tensor \[ \mathcal {R}(g)=\frac {1}{(\mathrm {Vol}(M))^{\frac {n-4}{n}}}\int _{M}\mid R \mid ^{2}dvol_{g} \] defines a Riemannian functional on the space of metrics. This work exhibits a metric on $S^3$ which is of Berger type but not of constant ricci curvature, and yet is critical for $\mathcal {R}$.