  We define geometric RG flow equations that specify the scale dependence of
the renormalized effective action Gamma[g] and the geometric entanglement
entropy S[x] of a QFT, considered as functionals of the background metric g and
the shape x of the entanglement surface. We show that for QFTs with AdS duals,
the respective flow equations are described by Ricci flow and mean curvature
flow. For holographic theories, the diffusion rate of the RG flow is much
larger, by a factor R_{AdS}^2/\ell_s^2, than the RG resolution length scale. To
derive our results. we employ the Hamilton-Jacobi equations that dictate the
dependence of the total bulk action and the minimal surface area on the
geometric QFT boundary data.
