A note on the Cotton flow and the Ricci flow for three-manifolds, and the Hořava–Lifshitz gravity

Abstract We consider the more general geometrical flow in the space of metrics for three-manifolds that consists of a combination of two flows, the Cotton flow and the Ricci flow; by playing a fundamental role in the detailed balance principle of the four dimensional Hořava–Lifshitz gravity, this generalized flow reveals another difficulty with this theory that attempts to be a candidate for an UV completion of Einstein general relativity, namely, the supposed emergency of the speed of light, the Newton constant, and the cosmological constant, from a deeply nonrelativistic theory of gravity. Respecting that principle, the generalized flow shows the proliferation of different limits of the theory with an unwanted behavior at both the IR and UV regimes.