论文标题
Rényi在Lorentzian空间上的熵。定时曲率维度条件
Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions
论文作者
论文摘要
对于以$ \ mathfrak {m} $衡量的Lorentzian空间，就Kunzinger，Sämann，Cavalletti和Mondino而言，我们介绍并研究了及时的Ricci曲率的合成概念，由$ k \ in \ boldsymbol in \ boldsymbol { $ n \在[1，\ infty）$中，即时机曲率尺寸条件$ \ smash {\ mathrm {\ mathrm {tcd} _p（k，n）} $和$ \ smash {\ smash {\ smash {\ mathrm {tcd} tcd} _p^*（k，k，n）$ in Beart and Strong and there $ p \ in $ p. MEATH-CONTRICTION属性$ \ SMASH {\ MATHRM {TMCP}（k，n）} $和$ \ SMASH {\ MATHRM {TMCP}^*（K，N）} $。这些是通过$ \ mathfrak {m} $沿$ \ smash {\ ell_p} $ - 概率度量的Geodesics沿$ \ mathfrak {m} $沿$ \ mathfrak {m} $的凸属性提出的。 我们展示了这些概念的许多特征，包括它们与光滑的设置，敏锐的几何不等式，稳定性，命名弱和强的版本的等价性，局部到全球特性的等效性以及年代学$ \ Smash {\ ell_p} $ - 最佳couplings and oftiral couplings and optiral couplings and-Chronologic $ \ smash smash od \ smash {$ geodes} $ geodes}我们还证明了$ \ smash {\ mathrm {tcd} _p^**（k，n）} $和$ \ smash {\ mathrm {tmcp}^*（k，n）} $的等效性。 其中一些结果是根据Timelike $ p $ - 必不分支的，这一概念比定时非分支的概念弱。
For a Lorentzian space measured by $\mathfrak{m}$ in the sense of Kunzinger, Sämann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by $K\in\boldsymbol{\mathrm{R}}$ and upper dimension bounds by $N\in[1,\infty)$, namely the timelike curvature-dimension conditions $\smash{\mathrm{TCD}_p(K,N)}$ and $\smash{\mathrm{TCD}_p^*(K,N)}$ in weak and strong forms, where $p\in (0,1)$, and the timelike measure-contraction properties $\smash{\mathrm{TMCP}(K,N)}$ and $\smash{\mathrm{TMCP}^*(K,N)}$. These are formulated by convexity properties of the Rényi entropy with respect to $\mathfrak{m}$ along $\smash{\ell_p}$-geodesics of probability measures. We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological $\smash{\ell_p}$-optimal couplings and chronological $\smash{\ell_p}$-geodesics. We also prove the equivalence of $\smash{\mathrm{TCD}_p^*(K,N)}$ and $\smash{\mathrm{TMCP}^*(K,N)}$ to their respective entropic counterparts in the sense of Cavalletti and Mondino. Some of these results are obtained under timelike $p$-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.
