A quantitative analysis of metrics on $\mathbf{R}^n$ with almost   constant positive scalar curvature, with applications to fast diffusion flows
  We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ 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 $\mathbf{R}^n$ related to the Yamabe flow. 
Giulio Ciraolo, Alessio Figalli, Francesco Maggi
math.AP math.MG