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

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 $Ω$ 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: \[ λ_1(Ω) - λ_1(B) \gtrsim |ΩΔB|^2 + \int |u_Ω - u_B|^2, \] where $B$ denotes the nearest geodesic ball to $Ω$...