Stability estimates for the sharp spectral gap bound under a curvature-dimension condition (Q6621683)

The authors study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. \N\NThe main result 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 \(L^1\)-functional inequality for RCD spaces with Stein's method for distribution approximation.\N\NThe authors also derive analogous, almost sharp, estimates for infinite and negative values of the dimension parameter.