The Willmore center of mass of initial data sets

TL;DR

This paper refines analytical methods to study the geometric center of mass of initial data sets in general relativity, revealing conditions under which it aligns with or differs from the Hamiltonian center of mass.

Contribution

It advances the understanding of the geometric center of mass in initial data sets by refining Lyapunov-Schmidt analysis and clarifying its relation to the Hamiltonian center of mass.

Findings

Geometric center of mass agrees with Hamiltonian center when scalar curvature vanishes at infinity.

Large area-constrained Willmore surfaces' position depends on energy density distribution.

Differences occur if scalar curvature lacks asymptotic symmetry.

Abstract

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.