  We refine the Lyapunov-Schmidt analysis developed in our recent paper
arxiv:2101.12665 to study the geometric center of mass of the asymptotic
foliation by area-constrained Willmore surfaces of initial data for the
Einstein field equations. If the scalar curvature of the initial data vanishes
at infinity, we show that this geometric center of mass agrees with the
Hamiltonian center of mass. By contrast, we show that the position of large
area-constrained Willmore surfaces is sensitive to the distribution of the
energy density. In particular, the geometric center of mass may differ from the
Hamiltonian center of mass if the scalar curvature does not satisfy asymptotic
symmetry assumptions.
