  We found Bogomol'nyi 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.
