  The objective of this paper is two-fold. First, we establish new sharp
quantitative estimates for Faber-Krahn inequalities on simply connected space
forms. We prove that the gap between the first eigenvalue of a given set
$\Omega$ and that of the ball quantitatively controls both the $L^1$ distance
of this set from a ball {\it and} the $L^2$ distance between the corresponding
eigenfunctions: \[ \lambda_1(\Omega) - \lambda_1(B) \gtrsim |\Omega \Delta B|^2
+ \int |u_{\Omega} - u_B|^2, \] where $B$ denotes the nearest geodesic ball to
$\Omega$ with $|B|=|\Omega|$ and $u_\Omega$ denotes the first eigenfunction
with suitable normalization. On Euclidean space, this extends a result of
Brasco-De Phillipis-Velichkov; the eigenfunction control largely builds upon
new regularity results for minimizers of critically perturbed Alt-Cafarelli
type functionals in our companion paper. On the round sphere and hyperbolic
space, the present results are the first sharp quantitative results with
respect to any distance; here the local portion of the analysis is based on new
implicit spectral analysis techniques.
  Second, we apply these sharp quantitative Faber-Krahn inequalities in order
to establish a quantitative form of the Alt-Caffarelli-Friedman (ACF)
monotonicity formula. We show that the energy drop in the ACF monotonicity
formula from one scale to the next controls how close a pair of admissible
functions is from a pair of complementary half-plane solutions. In particular,
when the square root of the energy drop summed over all scales is small, our
result implies the existence of tangents (unique blowups) of these functions.
