The Constraint Equations

TL;DR

This paper reviews the geometric origin, solution space, and construction techniques of the Einstein constraint equations, highlighting their role in the Cauchy problem and Hamiltonian formulation.

Contribution

It provides a comprehensive overview of the properties, solution spaces, and advanced construction methods for the Einstein constraint equations.

Findings

Analysis of the geometric origin of constraints

Overview of solution space properties

Discussion of advanced construction techniques

Abstract

We review the properties of the constraint equations, from their geometric origin in hypersurface geometry through to their roles in the Cauchy problem and the Hamiltonian formulation of the Einstein equations. We then review properties of the space of solutions and construction techniques, including the conformal and conformal thin sandwich methods, the thin sandwich method, quasi-spherical and generalized QS methods, gluing techniques and the Corvino-Schoen projection.