Heat Flow on Alexandrov Spaces
Heat Flow on Alexandrov Spaces
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DOI：
10.1002/cpa.21431
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发表时间：
2010-08
影响因子：
3
通讯作者：
N. Gigli;Kazumasa Kuwada;Shin-ichi Ohta
中科院分区：
文献类型：
--
作者：
N. Gigli;Kazumasa Kuwada;Shin-ichi Ohta
我们证明了在曲率有界于Dirichlet能量梯度流以下的紧Alexandrov空间上，空间产生的演化与文献类{文章}\usepackage{mathackage}\usepackage{amsath，amssymb}\usepackage{amsath，\egin{Document}\Begin{Align*}L^2\end{Align*}\end{Document}-空间中的相对熵的梯度流相同，Amssymb}\Page Style{Empty}\Begin{Document}\Begin{Align*}L^2\End{Align*}\End{Document}-Wasserstein空格。这意味着热流是由两个梯度流中的任何一个很好地定义的。结合这些流的性质，我们能够推导出热核的Lipschitz连续性，以及Bakry-Émery梯度估计和文献类{文章}\usepackage{mathsfs}\usepackage{amsath，amssymb}\Page Style{Empty}\Begin{Document}\Begin{Align*}\Gamma_2\End{Align*}\end{Document}-条件。与之前依赖于PDE技术的结果不同，我们的识别是通过纯粹的度量方法建立的。我们的方法推广到有漂移的热流的情况。©2012 Wiley期刊，Inc.
We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐space produces the same evolution as the gradient flow of the relative entropy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐Wasserstein space. This means that the heat flow is well‐defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry‐Émery gradient estimates and the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\Gamma_2\end{align*} \end{document}‐condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift. © 2012 Wiley Periodicals, Inc.