Quantitative Stability for Minimizing Yamabe Metrics

Abstract

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

23 pages. Minor edits and clarifications as suggested by the referee and others. Final version, accepted to Trans. AMS