STABILITY ESTIMATES FOR THE SHARP SPECTRAL GAP BOUND UNDER A CURVATURE-DIMENSION CONDITION

— We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an RCD(N − 1, N) space is almost minimal, then the pushforward of the measure by an eigenfunction associated with the spectral gap is close to a Beta distribution. The proof combines estimates on the eigenfunction obtained via a new L¹-functional inequality for RCD spaces with Stein’s method for distribution approximation. We also derive analogous, almost sharp, estimates for infinite and negative values of the dimension parameter.