Curvature homogeneous critical metrics in dimension three

Abstract

We study curvature homogeneous three-manifolds modeled on a symmetric space which are critical for some quadratic curvature functional. If the Ricci operator is diagonalizable, critical metrics are 1-curvature homogeneous Brinkmann waves and are critical for one specific functional. Otherwise, critical metrics are modeled on Cahen-Wallach symmetric spaces and they are Kundt spacetimes which are critical for all quadratic curvature functionals.