A canonical infinitesimally Hilbertian structure on locally Minkowski spaces

Abstract

The aim of this paper is to show the existence of a canonical distance ${\mathsf{d}}^{\prime }$ defined on a locally Minkowski metric measure space $(\mathsf{X},\mathsf{d},\mathfrak{m})$ such that: i) ${\mathsf{d}}^{\prime }$ is equivalent to $\mathsf{d}$, ii) $(\mathsf{X},{\mathsf{d}}^{\prime },\mathfrak{m})$ 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 ${\mathsf{d}}^{\prime }$ 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.