  The main objective of this paper is to control the geometry of a future
outgoing truncated null cone extending smoothly toward infinity in an
Einstein-vacuum spacetime. In particular, we wish to do this under minimal
regularity assumptions, namely, at the (weighted) L^2-curvature level. We show
that if the curvature flux and the data on an initial sphere of the cone are
sufficiently close to the corresponding values in a standard Minkowski or
Schwarzschild null cone, then we can obtain quantitative bounds on the geometry
of the entire infinite cone. The same bounds also imply the existence of limits
at infinity of the natural geometric quantities. Furthermore, we make no global
assumptions on the spacetime, as all assumptions are applied only to this
single truncated cone. In our sequel paper, we will apply these results in
order to control the Bondi energy and the angular momentum associated with this
cone.

Analysis of PDEs;General Relativity and Quantum Cosmology