标题： 关于合成零超曲面的几何学 显示英文标题

标题： On the geometry of synthetic null hypersurfaces

摘要： 本文提出了一种综合框架，用于非光滑时空中的零（光）超曲面的几何和分析研究。 结合最优传输理论以及洛伦兹几何和因果性理论的最新进展，我们将\emph{合成零曲面}定义为一个三元组$(H, G, \mathfrak{m})$：$H$是拓扑因果空间中的闭类空集，$G$是一个规范函数，用于编码沿零生成元的仿射参数化，而$\mathfrak{m}$是一个Radon测度，作为规度量的综合类似物。 这一方法将经典的微分几何结构推广到了可能具有奇点的时空中。 一个中心对象是\emph{合成零能量条件}($\mathsf{NC}^e(N)$)，它通过最优传输过程中熵功率泛函的凹性来定义，参数化由规范$G$给出。 此条件在自然等价类内的规范和测度变化下保持不变。 它在光滑情况下与经典的零能量条件一致，并适用于低正则性时空。 ($\mathsf{NC}^e(N)$) 的一个关键性质是合成零曲面在收敛下的稳定性，这受到测度 Gromov--Hausdorff 收敛的启发。 作为第一个应用，我们得到了霍金面积定理的合成版本。 此外，我们将著名的彭罗斯奇点定理扩展到连续时空，并证明了在满足合成 ($\mathsf{NC}^e(N)$) 的拓扑因果空间一般设定中存在被俘区域。 显示英文摘要

摘要： This paper develops a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a \emph{synthetic null hypersurface} as a triple $(H, G, \mathfrak{m})$: $H$ is a closed achronal set in a topological causal space, $G$ is a gauge function encoding affine parametrizations along null generators, and $\mathfrak{m}$ is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. A central object is the \emph{synthetic null energy condition} ($\mathsf{NC}^e(N)$), defined via the concavity of an entropy power functional along optimal transport, with parametrization given by the gauge $G$. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of ($\mathsf{NC}^e(N)$) is the stability under convergence of synthetic null hypersurfaces, inspired by measured Gromov--Hausdorff convergence. As a first application, we obtain a synthetic version of Hawking's area theorem. Moreover, we extend the celebrated Penrose's singularity theorem to continuous spacetimes and prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic ($\mathsf{NC}^e(N)$).