A canonical infinitesimally Hilbertian structure on locally Minkowski spaces

2022/03/17 by Mattia Magnabosco, Magnabosco, Mattia, Chiara Rigoni +1 · 1 citation
Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.2203.09643

openalex publication_date 2022/03/17 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

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_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_KS, \mathfrak m) is in fact the Korevaar-Schoen energy.

Cited by

• The infinitesimal structure of manifolds with non-continuous Riemannian metrics

Related

• The Hellinger-Kantorovich metric measure geometry on spaces of measures
• Universal infinitesimal Hilbertianity of sub-Riemannian manifolds
• Infinitesimal Hilbertianity of locally CAT(κ)-spaces
• The infinitesimal structure of manifolds with non-continuous Riemannian metrics
• Dirichlet forms on metric measure spaces as Mosco limits of Korevaar-Schoen energies