Ricci收缩度量测度空间上的热核
Heat kernel on Ricci shrinker metric measure spaces
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中文总结 AI 辅助
本文系统研究Ricci收缩度量测度空间上的热核，建立基本估计，阐明其与Ricci流时空热核的等价性，并推广Nash熵与约化距离，进而证明梯度估计蕴含曲率衰减。
中文摘要 AI 辅助
作为一个具有正Bakry-Émery曲率的度量测度空间，带势函数$f$的Ricci收缩子在加权体积测度$e^{-f}dv$下允许一个热核$H_f$。本文系统研究$H_f$，并建立一系列基本估计。此外，我们阐明了$H_f$与由Ricci收缩子诱导的Ricci流下的时空热核$H(x,t;y,s)$之间的等价性。受此关系启发，我们将Ricci流的相应分析工具（如Nash熵和约化距离）推广到Ricci收缩度量测度空间。作为直接应用，我们证明了$\left|\nabla R\right|=o(f^{3/2})$蕴含$R=o(f)$。
英文摘要
As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function $f$ admits a heat kernel $H_f$ under the weighted volume measure $e^{-f}dv$. In this paper, we study $H_f$ systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between $H_f$ and the spacetime heat kernel $H(x,t;y,s)$ under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that $\left|\nabla R\right|=o(f^{3/2})$ implies $R=o(f)$.
