A classical approach to the problem of effectively parametrizing the space of solutions to the vacuum Einstein constraint equations is the conformal method. We will focus on the constant mean curvature case, which reduces to the analysis of the Lichnerowicz equation to solve the Hamiltonian constraint, and which brings the study of the scalar curvature operator to the fore. We begin the chapter with elements of the theory of elliptic partial differential equations, and then develop the conformal method and present in detail the closed constant mean curvature case. In the last section we discuss more general scalar curvature deformation and an obstruction to positive scalar curvature due to Schoen and Yau.

This chapter develops the geometry of and analysis on initial data sets that arise in models of isolated gravitational systems. We begin with some detailed discussion and analysis involving the Laplace operator on asymptotically flat manifolds, which we use to develop density and deformation results on scalar curvature, leading to a proof of the Riemannian positive mass theorem. In the last section of the chapter we develop a technique for localized scalar curvature deformation, and we apply it to glue an asymptotically flat end with vanishing scalar curvature to an end of a Riemannian Schwarzschild metric, maintaining zero scalar curvature throughout.

Many physical models admit an initial value formulation. In this chapter we discuss an initial value formulation for the vacuum Einstein equation. A vacuum initial data set will be given geometrically as a manifold endowed with Riemannian metric and a symmetric two-tensor. That these give the first and second fundamental forms of an embedding into a Lorentzian manifold satisfying the vacuum Einstein equation imposes, via the Gauss and Codazzi equations, constraints on the initial data. These conditions, which govern the space of allowable initial data sets for the vacuum Einstein equation, comprise the Einstein constraint equations, the study of solutions to which form an interesting and rich subject for geometric analysis.