We introduce a description of a minimal surface in a space with boundary, as the world-hypersurface that the entangling surface traces. It does so by evolving from the boundary to the interior of the bulk under an appropriate geometric flow, whose parameter is the holographic coordinate. We specify this geometric flow for arbitrary bulk geometry. In the case of pure AdS spaces, we implement a perturbative approach for the solution of the flow equation around the boundary. We systematically study both the form of the perturbative solution as well as its dependence on the boundary conditions. This expansion is sufficient for the determination of all the divergent terms of the holographic entanglement entropy, including the logarithmic universal terms in odd spacetime bulk dimensions, for an arbitrary entangling surface, in terms of the extrinsic geometry of the latter.
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