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We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ thatare conformal to the flat metric, have almost constant positive scalarcurvature, and cannot concentrate more than one bubble. As an application ofour result, we show a quantitative rate of convergence in relative entropy fora fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.

A quantitative analysis of metrics on $\mathbf{R}^n$ with almost constant positive scalar curvature, with applications to fast diffusion flows