Geodesic metric spaces with unique blow-up almost everywhere: properties and examples

TL;DR

This paper explores metric spaces where almost every point has a unique tangent space, often a subFinsler Carnot group, highlighting properties and examples of such spaces.

Contribution

It characterizes conditions under which metric spaces have unique tangent spaces almost everywhere, including cases with Euclidean tangents despite pathological initial metrics.

Findings

Most points have tangent spaces that are subFinsler Carnot groups.

Some spaces have Euclidean tangents despite being pathological.

Conditions for uniqueness of tangent spaces are discussed.

Abstract

This short note has been written as an Oberwolfach report for the workshop "Differentialgeometrie im Grossen". We discuss properties of metric spaces that at almost all points admit a tangent metric space. We explain why, under some mild assumptions, the tangents are almost surely subFinsler Carnot groups. We mention several situations when the tangents are Euclidean spaces, but the initial metric spaces are quite pathological.