Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911149790855168 |
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| author | Núñez-Zimbrón, Jesús Pasqualetto, Enrico Soultanis, Elefterios |
| author_facet | Núñez-Zimbrón, Jesús Pasqualetto, Enrico Soultanis, Elefterios |
| contents | 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 $\mathbb{R}$), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian Núñez-Zimbrón, Jesús Pasqualetto, Enrico Soultanis, Elefterios Metric Geometry Differential Geometry Functional Analysis 46E36, 51F99, 49J52, 53C23 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 $\mathbb{R}$), 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. |
| title | Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian |
| topic | Metric Geometry Differential Geometry Functional Analysis 46E36, 51F99, 49J52, 53C23 |
| url | https://arxiv.org/abs/2508.05483 |