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. References

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a
conformal metric of constant scalar curvature (CSC). An affirmative answer was given by
Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing
the existence of a function that minimizes the so-called Yamabe energy functional; the
minimizing function corresponds to the conformal factor of the CSC metric.