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

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis 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

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.