  We prove lower large deviations for geometric functionals in sparse, critical
and dense regimes. Our results are tailored for functionals with nonexisting
exponential moments, for which standard large deviation theory is not
applicable. The primary tool of the proofs is a sprinkling technique that,
adapted to the considered functionals, ensures a certain boundedness. This
substantially generalizes previous approaches to tackle lower tails with
sprinkling. Applications include subgraph counts, persistent Betti numbers and
edge lengths based on a sparse random geometric graph, power-weighted edge
lengths of a $k$-nearest neighbor graph as well as power-weighted spherical
contact distances in a critical regime and volumes of $k$-nearest neighbor
balls in a dense regime.
