Sharp $L^2$-stability and gradient stability hierarchies of the $L^2$-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality

We study the sharp $L^2$-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space.