On one-dimensionality of metric measure spaces

TL;DR

This paper proves that certain metric measure spaces with open sets isometric to an interval and unique optimal transport maps are essentially one-dimensional manifolds, extending to various curvature-dimension conditions.

Contribution

It establishes that metric measure spaces with specific geometric and transport properties are necessarily one-dimensional manifolds, generalizing previous results under curvature and branching conditions.

Findings

Spaces with an open interval are one-dimensional manifolds.

Spaces with unique optimal transport maps are one-dimensional.

Results apply to $CD(K,N)$ and $MCP(K,N)$ spaces under certain conditions.

Abstract

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and for which the optimal transport maps not only exist but are unique. Again, we obtain an analogous corollary in the setting of essentially non-branching $MCP(K,N)$-spaces.