Towards a proof of a rigidity conjecture for asymptotically flat spacetimes

I will discuss ongoing work to provide a proof for the following

conjecture: if the development of a time symmetric, conformally flat

initial data set admits a smooth null infinity, then the initial data

is Schwarzschildean in a neighbourhood of infinity. The strategy

to construct a proof consists in a detailed analysis of a

certain type of expansions that can be obtained using H. Friedrich's

"cylinder at infinity" formalism. I will also discuss a toy model for

the analysis of the Maxwell field near the

spatial infinity of the Schwarzschild spacetime