  On any closed Riemannian manifold of dimension $n\geq 3$, 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.
