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

Abstract

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.

arXiv commentBecause of a gap in one proof in the previous version, we replaced the paper with a weaker version of our Theorem 1.1. Still, this result suffices to obtain the main conclusion in the previous paper, namely a quantitative convergence estimate for a fast diffusion equation related to the Yamabe flow