Fine Representation of Hessian of Convex Functions and Ricci Tensor on RCD Spaces

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-02 DOI:10.1007/s11118-024-10153-5

Camillo Brena, Nicola Gigli

引用次数: 0

Abstract

It is known that on RCD spaces one can define a distributional Ricci tensor $\textbf{Ric}$. Here we give a fine description of this object by showing that it admits the polar decomposition

$$\begin{aligned} \textbf{Ric}=\omega \,|\textbf{Ric}| \end{aligned}$$

for a suitable non-negative measure $|\textbf{Ric}|$ and unitary tensor field $\omega $. The regularity of both the mass measure and of the polar vector are also described. The representation provided here allows to answer some open problems about the structure of the Ricci tensor in such singular setting. Our discussion also covers the case of Hessians of convex functions and, under suitable assumptions on the base space, of the Sectional curvature operator.

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凸函数和里奇张量在 RCD 空间上的精细表示

众所周知，在 RCD 空间上，我们可以定义一个分布式里奇张量（\textbf{Ric}\）。在这里，我们通过证明它允许极性分解 $$\begin{aligned}，给出了这个对象的精细描述。\textbf{Ric}=\omega\,|\textbf{Ric}|\end{aligned}$$对于合适的非负度量$|\textbf{Ric}|$和单元张量场$\omega $。质量度量和极向量的正则性也得到了描述。这里提供的表示法可以回答在这种奇异设置下关于里奇张量结构的一些未决问题。我们的讨论还涉及凸函数的赫西亚，以及在基空间的适当假设下的截面曲率算子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.