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# 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

#### 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

Published 7 July 2015