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We refine the Lyapunov-Schmidt analysis developed in our recent paperarxiv:2101.12665 to study the geometric center of mass of the asymptoticfoliation by area-constrained Willmore surfaces of initial data for theEinstein field equations. If the scalar curvature of the initial data vanishesat infinity, we show that this geometric center of mass agrees with theHamiltonian center of mass. By contrast, we show that the position of largearea-constrained Willmore surfaces is sensitive to the distribution of theenergy density. In particular, the geometric center of mass may differ from theHamiltonian center of mass if the scalar curvature does not satisfy asymptoticsymmetry assumptions.

The Willmore center of mass of initial data sets