Local foliations by critical surfaces of the Hawking energy and small sphere limit

Abstract

Local foliations of area constrained Willmore surfaces on a 3-dimensional Riemannian manifold were constructed by Lamm et al (2020 Ann. Inst. Fourier 70 1639–62) and Ikoma et al (2020 Int. Math. Res. Not. 70 6538–68), 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 an 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 (2020 arXiv:1909.02388v2 [math.DG]), 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 (1982 Proc. R. Soc. A 381 215–24), 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 et al (2011 Math. Ann. 350 1–78), and we find an indication that these surfaces may induce an excess in the energy measured.