Shape dependence and RG flow of entanglement entropy

I will describe the results of two recent papers on entanglement entropy (EE). In the first paper we study the shape dependence of EE. We start with rotational/translational symmetric entangling surfaces and slightly deform them, and we compute the resulting corrections to the EE. These symmetric surfaces extremize the universal EE. In the second paper we study EE on the d-sphere, when the entangling surface is a (d-2)-sphere. In this setup the modular Hamiltonian and EE can be computed analytically in terms of the VEV of the energy-momentum tensor. We discuss applications to RG flows and c and F-theorems in d dimensions.