标题： 广义锥体容许曲率-维数条件 显示英文标题

标题： Generalized cones admitting a curvature-dimension condition

摘要： 我们研究了度量空间上的（广义）锥体，既包括黎曼情形也包括洛伦兹情形。 特别是，我们在度量测度的情形下建立了类似于 Lott-Villani-Sturm 和 Ohta 的合成下 Ricci 曲率界，以及在洛伦兹情形下类似于 Cavalletti-Mondino 的合成下 Ricci 曲率界。 这里，广义锥体是一维底空间（正定或负定）与纤维（一个度量空间）的扭曲乘积。 我们证明了： 如果黎曼或洛伦兹广义锥体是 $\mathsf{CD}$-空间，则满足（类时）测度收缩性质 $\mathsf{(T)MCP}$ —— 这是（类时）曲率-维数条件 $\mathsf{(T)CD}$ 的弱形式。 反之，如果广义锥体是 $\mathsf{(T)CD}$-空间，则纤维是具有适当 Ricci 曲率和维数限制的 $\mathsf{CD}$-空间。 在证明这些结果的过程中，我们发展了一种新颖且强大的二维局部化技术，我们预计这一技术本身会很有趣，并在其他情况下也有用。 最后，我们给出了几个应用，包括广义锥体的合成奇点和分裂定理。 最后一个应用是：我们通过给定空间上的广义锥体的下曲率界，提出了一种新的度量空间和度量测度空间的下曲率界的定义。 显示英文摘要

摘要： We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds \`a la Lott-Villani-Sturm and Ohta in the metric measure case, and \`a la Cavalletti-Mondino in Lorentzian signature. Here, a generalized cone is a warped product of a one-dimensional base space, which will be positive or negative definite, over a fiber that is a metric space. We prove that Riemannian or Lorentzian generalized cones over $\mathsf{CD}$-spaces satisfy the (timelike) measure contraction property $\mathsf{(T)MCP}$ - a weaker version of a (timelike) curvature-dimension condition $\mathsf{(T)CD}$. Conversely, if the generalized cone is a $\mathsf{(T)CD}$-space, then the fiber is a $\mathsf{CD}$-space with the appropriate bounds on Ricci curvature and dimension. In proving these results we develop a novel and powerful two-dimensional localization technique, which we expect to be interesting in its own right and useful in other circumstances. We conclude by giving several applications including synthetic singularity and splitting theorems for generalized cones. The final application is that we propose a new definition for lower curvature bounds for metric and metric measure spaces via lower curvature bounds for generalized cones over the given space.