A Quantitative Analysis of Metrics on R-n with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows

We prove a quantitative structure theorem for metrics on 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 R^n related to the Yamabe flow.