Curvature based triangulation of metric measure spaces

TL;DR

This paper extends Ricci curvature-based triangulation methods from Riemannian manifolds to weighted and general metric measure spaces, introducing improvements and applications including quasimeromorphic mappings and volume growth analysis.

Contribution

It generalizes triangulation techniques to weighted and metric measure spaces using the curvature-dimension condition, and explores applications like mappings and volume growth.

Findings

Triangulation extends to weighted Riemannian manifolds and metric measure spaces.

Weighted manifolds admit thick triangulations and weight-sensitive mappings.

Volume growth matches that of discretizations in weak CD(K,N) spaces.

Abstract

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition ${\rm CD}(K,N)$. We show also that for weighted Riemannian manifolds the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasimeromorphic mappings. An application of this last result to information manifolds is considered. Further more, we extend to weak ${\rm CD}(K,N)$ spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.