Extremal black holes and holographic c-theorem

We found Bogomolʼnyi type of the first order differential equations in three-dimensional Einstein gravity and the effective second order ones in new massive gravity when an interacting scalar field is minimally coupled. Using these equations in Einstein gravity, we obtain analytic solutions corresponding to extremally rotating hairy black holes. We also obtain perturbatively extremal black hole solutions in new massive gravity using these lower order differential equations. All these solutions have the anti-de Sitter spaces as their asymptotic geometries and as the near horizon ones. This feature of solutions interpolating two anti-de Sitter spaces leads to the construction of holographic c-theorem in these cases. Since our lower order equations reduce naturally to the well-known equations for domain walls, our results can be regarded as the natural extension of domain walls to more generic cases.