There have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. This talk will begin with introducing several results obtained by this foliation method. I will talk about my work on a particular construction of asymptotically hyperbolic Riemannian 3-manifolds using the Ricci flow solution on a closed surface as a foliation. This can be used to get an asymptotically hyperbolic extension from a closed surface with some condition.