Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 3

Special Issue: In Honor of Robert Bartnik (Part 2 of 2)

Guest Editors: Piotr T. Chruściel, Greg Galloway, Jim Isenberg, Pengzi Miao, Mu-Tao Wang, and Shing-Tung Yau

A perturbative approach to the construction of initial data on compact manifolds

Pages: 785 – 826

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n3.a1

Authors

Juan A. Valiente Kroon (School of Mathematical Sciences, Queen Mary, University of London, United Kingdom)

Jarrod L. Williams (School of Mathematical Sciences, Queen Mary, University of London, United Kingdom)

Abstract

We discuss the implementation, on compact manifolds, of the perturbative method of Friedrich–Butscher for the construction of solutions to the vacuum Einstein constraint equations. This method is of a perturbative nature and exploits the properties of the extended constraint equations—a larger system of equations whose solutions imply a solution to the Einstein constraints. The method is applied to the construction of nonlinear perturbations of constant mean curvature initial data of constant negative sectional curvature. We prove the existence of a neighbourhood of solutions to the constraint equations around such initial data, with particular components of the extrinsic curvature and electric/magnetic parts of the spacetime Weyl curvature prescribed as free data. The space of such free data is parametrised explicitly.

Keywords

Einstein constraint equations, elliptic systems, compact manifolds

2010 Mathematics Subject Classification

Primary 35Jxx, 83Cxx. Secondary 35Qxx.

Full Text (PDF format)

The authors thank the hospitality of the International Erwin Schrödinger Institute for Mathematics and Physics where a big part of this work was carried out as part of the research programme Geometry and Relativity during July-September 2017.

Received 8 May 2019

Received revised 23 July 2019

Accepted 11 August 2019

Published 2 January 2020