Generalized cones admitting a curvature-dimension condition

TL;DR

This paper investigates generalized cones over metric spaces in both Riemannian and Lorentzian signatures, establishing synthetic lower Ricci curvature bounds and developing a new localization technique with applications to singularity and splitting theorems.

Contribution

It introduces a novel two-dimensional localization method and links curvature-dimension conditions of cones to those of their fibers, advancing synthetic Ricci curvature theory.

Findings

Generalized cones over CD-spaces satisfy the TMCP property.

If a cone satisfies TCD, its fiber is a CD-space with corresponding bounds.

The new localization technique is effective for analyzing curvature conditions.

Abstract

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.