The Hawking energy is one of the most famous local energies in general relativity, but it has the inconvenience that it is highly depedend on the surface taken.To remedy this Lamm, Metzger and Schulze proposed to consider area constrained surfaces of the Hawking energy. By using a Lyapunov Schmidt reduction procedure we construct unique local foliations of critical surfaces of the Hawking energy on initial data sets (spacelike hypersurfaces in a spacetime), and we show a nonexistence condition. Any quasilocal energy should satisfy the so called small sphere limit, therefore we also discuss the relation of these surfaces and the small sphere limit. In particular we discuss some discrepancies on the small sphere limit, so when approaching a point with these foliations and when approaching as in the small sphere limit. We also find an indication that these surfaces may induce an excess in the energy measured
