ADM reduction of Einstein action and black hole entropy

We reduce the 4D Einstein-Hilbert action to a constant-radius hypersurface of foliation. The resulting theory is a scalar theory defined on a 3D hypersurface of the original black hole background, and has an exponential potential. Once the the hypersurface is located at the Schwarzschild radius, the 3D theory is effectively reduced to a 2D Liouville type theory. We compute {the entropy associated with the hypersurface intrinsic degrees of freedom}, and show that its leading order reproduces the Bekenstein-Hawking area law. The subleading terms come in logarithm/inverse powers of the area.