English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a Lorentzian space measured by $\mathfrak{m}$ in the sense of Kunzinger,S\"amann, Cavalletti, and Mondino, we introduce and study synthetic notions oftimelike lower Ricci curvature bounds by $K\in\boldsymbol{\mathrm{R}}$ andupper dimension bounds by $N\in[1,\infty)$, namely the timelikecurvature-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 convexityproperties of the R\'enyi entropy with respect to $\mathfrak{m}$ along$\smash{\ell_p}$-geodesics of probability measures.  We show many features of these notions, including their compatibility withthe smooth setting, sharp geometric inequalities, stability, equivalence of thenamed weak and strong versions, local-to-global properties, and uniqueness ofchronological $\smash{\ell_p}$-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 theirrespective 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.

R\'enyi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions