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The objective of this paper is two-fold. First, we establish new sharpquantitative estimates for Faber-Krahn inequalities on simply connected spaceforms. We prove that the gap between the first eigenvalue of a given set$\Omega$ and that of the ball quantitatively controls both the $L^1$ distanceof this set from a ball {\it and} the $L^2$ distance between the correspondingeigenfunctions: \[ \lambda_1(\Omega) - \lambda_1(B) \gtrsim |\Omega \Delta B|^2+ \int |u_{\Omega} - u_B|^2, \] where $B$ denotes the nearest geodesic ball to$\Omega$ with $|B|=|\Omega|$ and $u_\Omega$ denotes the first eigenfunctionwith suitable normalization. On Euclidean space, this extends a result ofBrasco-De Phillipis-Velichkov; the eigenfunction control largely builds uponnew regularity results for minimizers of critically perturbed Alt-Cafarellitype functionals in our companion paper. On the round sphere and hyperbolicspace, the present results are the first sharp quantitative results withrespect to any distance; here the local portion of the analysis is based on newimplicit spectral analysis techniques.  Second, we apply these sharp quantitative Faber-Krahn inequalities in orderto establish a quantitative form of the Alt-Caffarelli-Friedman (ACF)monotonicity formula. We show that the energy drop in the ACF monotonicityformula from one scale to the next controls how close a pair of admissiblefunctions 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, ourresult implies the existence of tangents (unique blowups) of these functions.

Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula