Chapter 8 - On the center of mass and constant mean curvature surfaces of asymptotically flat initial data sets

Summary

Many deep results in mathematical general relativity concern the interplay between globally conserved quantities, such as the center of mass and angular momentum, and the geometric structure of initial data sets, using analysis of the scalar curvature, or more generally the full constraint equations. The chapter focuses on constant mean curvature foliations and the geometric center of mass of asymptotically flat initial data sets, a research program initiated by Huisken and Yau. We begin with a partial survey of the classical results of constant mean curvature surfaces and introduce the concept of stability, and then discuss some recent progress on the constant mean curvature surfaces in asymptotically flat initial data sets and the geometric center of mass. In the final section, we adopt a more analytic approach to study the center of mass and angular momentum from the Einstein constraint equations.