On the Shape Dependence of Entanglement Entropy

We study the shape dependence of entanglement entropy (EE) by deforming symmetric entangling surfaces. We show that entangling surfaces with a rotational or translational symmetry extremize (locally) the EE with respect to shape deformations that break some of the symmetry (i.e. the 1st order correction vanishes). This result applies to EE and Renyi entropy for any QFT in any dimension. Using Solodukhin's formula in $4d$ and holography in any $d$, we calculate the 2nd order correction to the universal EE for CFTs and simple symmetric entangling surfaces. In all cases we find that the 2nd order correction is positive, and thus the corresponding symmetric entangling surface is a local minimum. Some of the results are extended to free massive fields and to 4d Renyi entropy.