标题： Hausdorff维数和亚洛伦兹Heisenberg群中合成曲率界限的失效 显示英文标题

标题： Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group

摘要： 我们研究了子洛伦兹海森堡群的测地线、豪斯多夫维数和曲率界限。 通过一种基本的变分方法，我们提供了其最大测地线结构的新证明，表明它们是来自闵可夫斯基平面上洛伦兹等周问题的双曲线的提升。 我们证明了该空间的洛伦兹豪斯多夫维数为$4$，并且相应的测度与哈尔测度一致。 我们进一步建立了一个类似于球盒定理的新结果，通过各向异性方块对因果钻石给出了统一估计。 最后，我们表明海森堡群对于参数$K$和$N$的任何值都不满足 timelike 曲率-维数条件$\mathsf{TCD}(K,N)$或 timelike 测度收缩性质$\mathsf{TMCP}(K,N)$，这与其子黎曼对应物形成鲜明对比。 显示英文摘要

摘要： We study the geodesics, Hausdorff dimension, and curvature bounds of the sub-Lorentzian Heisenberg group. Through an elementary variational approach, we provide a new proof of the structure of its maximizing geodesics, showing that they are lifts of hyperbolae coming from a Lorentzian isoperimetric problem in the Minkowski plane. We prove that the Lorentzian Hausdorff dimension of the space is $4$ and that the corresponding measure coincides with the Haar measure. We further establish a novel result in the spirit of the Ball-Box theorem, giving a uniform estimate of causal diamonds by anisotropic boxes. Finally, we show that the Heisenberg group satisfies neither the timelike curvature-dimension condition $\mathsf{TCD}(K,N)$ nor the timelike measure contraction property $\mathsf{TMCP}(K,N)$ for any values of the parameters $K$ and $N$, in sharp contrast with its sub-Riemannian counterpart.