  Here we prove a global existence theorem for the solutions of the semi-linear
wave equation with critical non-linearity admitting a positive definite
Hamiltonian. Formulating a parametrix for the wave equation in a globally
hyperbolic curved spacetime, we derive an apriori pointwise bound for the
solution of the nonlinear wave equation in terms of the initial energy, from
which the global existence follows in a straightforward way. This is
accomplished in two steps. First, based on Moncrief's light cone formulation we
derive an expression for the scalar field in terms of integrals over the past
light cone from an arbitrary spacetime point to an `initial', Cauchy
hypersurface and additional integrals over the intersection of this cone with
the initial hypersurface. Secondly, we obtain apriori estimates for the energy
associated with three quasi-local approximate time-like conformal Killing and
one approximate Killing vector fields. Utilizing these naturally defined
energies associated with the physical stress-energy tensor together with the
integral equation, we show that the spacetime $L^{\infty}$ norm of the scalar
field remains bounded in terms of the initial data and continues to be so as
long as the spacetime remains singularity/Cauchy-horizon free.
