A canonical infinitesimally Hilbertian structure on locally Minkowski spaces

TL;DR

This paper constructs a canonical metric on locally Minkowski spaces that makes them infinitesimally Hilbertian, broadening the class of spaces with such structures and linking the Korevaar-Schoen energy to Cheeger energy.

Contribution

It introduces a canonical distance making locally Minkowski spaces infinitesimally Hilbertian, expanding the class of such metric measure spaces beyond Finsler manifolds.

Findings

The new distance is equivalent to the original metric.

The space becomes infinitesimally Hilbertian with the new distance.

The Korevaar-Schoen energy is quadratic and coincides with the Cheeger energy.

Abstract

The aim of this paper is to show the existence of a canonical distance $\mathsf d'$ defined on a locally Minkowski metric measure space $(\mathsf X,\mathsf d,\mathfrak m)$ such that: i) $\mathsf d'$ is equivalent to $\mathsf d$, ii) $(\mathsf X, \mathsf d', \mathfrak m)$ is infinitesimally Hilbertian. This new regularity assumption on $(\mathsf X, \mathsf d,\mathfrak m)$ essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the Finsler manifolds, and it is actually strictly larger. The required distance $\mathsf d'$ will be the intrinsic distance $\mathsf d_\mathsf{KS}$ associated to the so-called Korevaar-Schoen energy, which is proven to be a quadratic form. In particular, we show that the Cheeger energy associated to the metric measure space $(\mathsf X, \mathsf d_\mathsf{KS}, \mathfrak m)$ is in fact the Korevaar-Schoen energy.