Inserted: 5 feb 2016
Last Updated: 22 mar 2017
Journal: Int. Math. Res. Not. IMRN
We prove a quantitative structure theorem for metrics on $\mathbb R^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbb R^n$ related to the Yamabe flow.