Surveys in Differential Geometry

Volume 20 (2015)

Conserved quantities of harmonic asymptotic initial data sets

Pages: 227 – 248

DOI: https://dx.doi.org/10.4310/SDG.2015.v20.n1.a9

Authors

Po-Ning Chen (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Mu-Tao Wang (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Abstract

In the first half of this article, we survey the new notions of quasi-local and total angular momentum and center of mass defined in [9] (P.-N. Chen, M.-T. Wang, and S.-T. Yau, Conserved quantities in general relativity: from the quasi-local level to spatial infinity), and summarize their important properties. The computation of these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is difficult in general. We found a large family of initial data sets on which such a calculation can be carried out effectively. These are initial data sets of harmonic asymptotics, first proposed by Corvino and Schoen. In the second half of this article, the new total angular momentum and center of mass for these initial data sets are computed explicitly.

Keywords

harmonic asymptotic initial data sets, conserved quantities, symmetry, mass, energy-momentum

2010 Mathematics Subject Classification

35Q76, 83C05, 83C30

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Published 7 July 2015