A quantitative analysis of metrics on $\R^n$ with almost constant positive scalar curvature, with applications to Yamabe and fast diffusion flows
A quantitative analysis of metrics on $\R^n$ with almost constant positive scalar curvature, with applications to Yamabe and fast diffusion flows
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对具有几乎恒定的正标量曲率的 $R^n$ 的度量进行定量分析，并应用于 Yamabe 和快速扩散流
DOI：
10.1093/imrn/rnx071
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
F. Maggi
中科院分区：
文献类型：
--
作者：
Giulio Ciraolo;A. Figalli;F. Maggi
我们证明了关于 $\R^n$ 度量结构的斯特鲁威定理的定量版本，该定理与平坦度量共形并且具有几乎恒定的正标量曲率。作为我们结果的两个应用，我们通过熵-熵产生不等式和 Bianchi-Egnell 改进的 Sobolev 不等式证明了球体上保体积 Yamabe 流的收敛性，并且我们展示了与 Yamabe 流相关的 $\R^n$ 中快速扩散方程的相对熵的定量收敛率。
We prove a quantitative version of Struwe's theorem on the structure of metrics on $\R^n$ which are conformal to the flat metric and have almost constant positive scalar curvature. As two applications of our result, we prove the convergence of the volume-preserving Yamabe flow on the sphere by means of a entropy-entropy production inequality and of the improved Sobolev inequality by Bianchi-Egnell, and we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\R^n$ related to the Yamabe flow.