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Generalized cones admitting a curvature-dimension condition

Matteo Calisti, Christian Ketterer, Clemens Sämann

Abstract

We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds à la Lott-Villani-Sturm and Ohta in the metric measure case, and à 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.

Generalized cones admitting a curvature-dimension condition

Abstract

We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds à la Lott-Villani-Sturm and Ohta in the metric measure case, and à 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 -spaces satisfy the (timelike) measure contraction property - a weaker version of a (timelike) curvature-dimension condition . Conversely, if the generalized cone is a -space, then the fiber is a -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.

Paper Structure

This paper contains 28 sections, 43 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal transport and curvature-dimension conditions
  4. Curvature-dimension conditions for metric measure spaces
  5. Lorentzian (pre-)length spaces
  6. Timelike curvature-dimensions conditions for Lorentzian spaces
  7. Weighted Lorentzian manifolds and the Bakry--Émery condition
  8. The localisation method on metric measure spaces
  9. Disintegration of measures
  10. $1D$ localisation
  11. Characterization of curvature bounds via $1D$ localisation
  12. Generalized cones and warped products
  13. Warped products over semi-Riemannian manifolds
  14. $1$-dimensional base and fiber
  15. Generalized cones as metric spaces
  16. ...and 13 more sections

Key Result

Theorem 4.2

Let $N>1$ and assume $\partial I= f^{-1}(\{0\})$. If $h\in C^0([a,b], [0, \infty))$ satisfies in the distributional sense, then the generalized cone $({}^\pm I\times_f [a,b], {f^{N-1}h}\,\mathrm{vol}_{^\pm I\times_f [a,b]})$ satisfies $\textnormal{CD}(\kappa N, N+1)$ or $\textnormal{TCD}_p(-\kappa N,N+1)$, respectively. Here $\mathrm{vol}_{^\pm I\times_f [a,b]}$ is the volume measure $\sqrt{

Theorems & Definitions (113)

  • Theorem 4.2
  • Theorem 4.10
  • Theorem 5.2: From $X$ to ${}^-\!I\times^N_fX$
  • Theorem 5.6: From $X$ to ${}^+I\times^N_fX$
  • Remark 1.1
  • Theorem 6.1: From ${}^-\!I\times^N_fX$ to $X$
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1: (Essential) nonbranching
  • Definition 2.2: $\textnormal{CD}(K,\infty)$
  • ...and 103 more