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Geometric RG Flow

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.