On the gravitational Chern-Simons action as entropy functional for three-manifolds, and the demystification of the Hov{r}ava-Lifshitz gravity

We determine the more general geometrical flow in the space of metrics corresponding to the steepest descent for the three-dimensional gravitational Chern-Simons action, extending the results previously considered in Class. Quantum Grav. 25 (2008) 165019, and reveling another trouble with the four dimensional Ho\v{r}ava-Lifshitz gravity introduced in Phys. Rev. D 79, 084008 (2009), and JHEP 0903, 020 (2009), which attempts to be a candidate for an UV completion of Einstein general relativity.