  Local foliations of area constrained Willmore surfaces on a 3-dimensional
Riemannian manifold were constructed by Lamm, Metzger and Schulze, and Ikoma,
Machiodi and Mondino, the leaves of these foliations are, in particular,
critical surfaces of the Hawking energy in case they are contained in a totally
geodesic spacelike hypersurface. We generalize these foliations to the general
case of a non-totally geodesic spacelike hypersurface, constructing a unique
local foliation of area constrained critical surfaces of the Hawking energy. A
discrepancy when evaluating the so called small sphere limit of the Hawking
energy was found by Friedrich. He studied concentrations of area constrained
critical surfaces of the Hawking energy and obtained a result that apparently
differs from the well established small sphere limit of the Hawking energy of
Horowitz and Schmidt, this small sphere limit in principle must be satisfied by
any quasi local energy. We independently confirm the discrepancy and explain
the reasons for it to happen. We also prove that these surfaces are suitable to
evaluate the Hawking energy in the sense of Lamm, Metzger and Schulze, and we
find an indication that these surfaces may induce an excess in the energy
measured.
