Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds

The Hamiltonian constraint ``G00 = 8πT00'' of general relativity is written as a quasilinear elliptic differential equation for the conformal factor of the metric of a three-dimensional spacelike manifold. It is shown that for ``almost every'' configuration of initial data on a compact manifold, with or without boundary, a solution exists. Dirichlet boundary conditions are assumed if the boundary is not empty. The solution is unique.