Quantitative stability for minimizing Yamabe metrics

On any closed Riemannian manifold of dimension n ≥ 3 n\geq 3 , we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false.