  We study the finite term of the holographic entanglement entropy of finite
domains with smooth shapes and for four dimensional gravitational backgrounds.
Analytic expressions depending on the unit vectors normal to the minimal area
surface are obtained for both stationary and time dependent spacetimes. The
special cases of AdS4, asymptotically AdS4 black holes, domain wall geometries
and Vaidya-AdS backgrounds have been analysed explicitly. When the bulk
spacetime is AdS4, the finite term is the Willmore energy of the minimal area
surface viewed as a submanifold of the three dimensional flat Euclidean space.
For the static spacetimes, some numerical checks involving spatial regions
delimited by ellipses and non convex domains have been performed. In the case
of AdS4, the infinite wedge has been also considered, recovering the known
analytic formula for the coefficient of the logarithmic divergence.
