  Impulsive gravitational waves are (weak) solutions to the Einstein vacuum
equations such that the Riemann curvature tensor admits a delta singularity
along a null hypersurface. The interaction of impulsive gravitational waves is
then represented by the transversal intersection of these singular null
hypersurfaces.
  This is the first of a series of two papers in which we prove that for all
suitable $\mathbb U(1)$-symmetric initial data representing three "small
amplitude" impulsive gravitational waves propagating towards each other
transversally, there exists a local solution to the Einstein vacuum equations
featuring the interaction of these waves. Moreover, we show that the solution
remains Lipschitz everywhere and is $H^2_{loc} \cap C_{loc}^{1, \frac 14-}$
away from the impulsive gravitational waves. This is the first construction of
solutions to the Einstein vacuum equations featuring the interaction of three
impulsive gravitational waves.
  In this paper, we focus on the geometric estimates, i.e. we control the
metric and the null hypersurfaces assuming the wave estimates. The geometric
estimates rely crucially on the features of the spacetime with three
interacting impulsive gravitational waves, particularly that each wave is
highly localized and that the waves are transversal to each other. In the
second paper of the series, we will prove the wave estimates and complete the
proof.
