On the Geometry of Null Cones to Infinity Under Curvature Flux Bounds

TL;DR

This paper establishes minimal-regularity geometric bounds for outgoing null cones in Einstein-vacuum spacetimes, ensuring control over their infinity limits under curvature flux bounds.

Contribution

It introduces a method to bound the geometry of null cones under minimal regularity assumptions, extending control to infinity without global spacetime assumptions.

Findings

Quantitative bounds on null cone geometry under curvature flux conditions

Existence of limits at infinity for geometric quantities

Applicability to control Bondi energy and angular momentum in future work

Abstract

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.