Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian

2025/08/07 by Núñez-Zimbrón, Jesús, Pasqualetto, Enrico, Soultanis, Elefterios · 1 citation
#46E36 #49J52 #51F99 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.05483

Abstract

We show that a metric space X that, at every point, has a Gromov-Hausdorff tangent with the splitting property (i.e. every geodesic line splits off a factor ℝ), is universally infinitesimally Hilbertian (i.e. W^1,2(X,μ) is a Hilbert space for every measure μ). This connects the infinitesimal geometry of X to its analytic properties and is, to our knowledge, the first general criterion guaranteeing universal infinitesimal Hilbertianity. Using it we establish universal infinitesimal Hilbertianity of finite dimensional RCD-spaces. We moreover show that (possibly infinite dimensional) Alexandrov spaces are universally infinitesimally Hilbertian and construct an isometric embedding of tangent modules.

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Cited by

• Inradius collapsed manifolds with a lower Ricci curvature bound

Related

• Infinitesimal Hilbertianity of locally CAT(κ)-spaces
• The splitting theorem in non-smooth context
• Universal infinitesimal Hilbertianity of sub-Riemannian manifolds
• The infinitesimal structure of manifolds with non-continuous Riemannian metrics
• Non-Hilbertian tangents to Hilbertian spaces