类时Ollivier-Ricci曲率
Timelike Ollivier-Ricci curvature
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中文总结 AI 辅助
本文在洛伦兹几何中引入余维数为1的粗Ricci曲率构造，通过1-Lorentz-Wasserstein距离和Moser型流恢复类时方向上的Ricci曲率及Bakry-Émery张量，呼应Raychaudhuri方程。
中文摘要 AI 辅助
我们在洛伦兹几何中引入了余维数为1的粗Ricci曲率构造。利用$1$-Lorentz-Wasserstein距离$\ell_1$，我们通过邻近事件比较支撑在小类空超曲面上的概率测度，并在精确的渐近区域中恢复未来方向单位类时方向上的环境Ricci曲率，其中包含反映该余维数构造的普适维度前因子。该构造呼应了Raychaudhuri方程，该方程将沿类时测地线的空间体积膨胀演化与Ricci曲率联系起来。我们的定量估计依赖于通过Moser型流构造的具有受控位移的传输映射。在领头阶，切片构造对光滑权重不敏感。通过将类空切片涂抹成细类时管并校准时间与空间尺度，我们恢复了与加权参考测度$\smash{\mathfrak{m}=\mathrm{e}^{-V}\\,\mathrm{vol}_g}$相关联的类时Bakry-Émery张量$\mathrm{Ric}+\mathrm{Hess}\\,V$。
英文摘要
We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance $\ell_1$, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor $\mathrm{Ric}+\mathrm{Hess}\,V$ associated with the weighted reference measure $\smash{\mathfrak{m}=\mathrm{e}^{-V}\,\mathrm{vol}_g}$.
发表机构
- EPFL（洛桑联邦理工学院）
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