Local-to-global Urysohn width estimates
-
Alexey Balitskiy
und
Aleksandr Berdnikov
Abstract
The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable
Acknowledgements
We are grateful to Larry Guth for numerous conversations and his remarks on this paper. We also thank Hannah Alpert and Panos Papasoglu for the stimulating discussions that led us to these questions. The results constitute a chapter of the thesis defended by the first-named author under the supervision of Larry Guth at MIT.
References
[1] A. Akopyan, R. Karasev and A. Volovikov, Borsuk–Ulam type theorems for metric spaces, preprint (2012), https://arxiv.org/abs/1209.1249. Suche in Google Scholar
[2] P. Alexandroff, Notes supplémentaires au “Mémoire sur les multiplicités Cantoriennes”, rédigées d’après les papiers posthumes de Paul Urysohn, Fund. Math. 1 (1926), no. 8, 352–359. 10.4064/fm-8-1-352-359Suche in Google Scholar
[3] L. E. J. Brouwer, Über den natürlichen Dimensionsbegriff, J. reine angew. Math. 142 (1913), 146–152. 10.1515/crll.1913.142.146Suche in Google Scholar
[4] M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. 10.4310/jdg/1214509283Suche in Google Scholar
[5] M. Gromov, Width and related invariants of Riemannian manifolds, Astérisque 163–164 (1988), 93–109. Suche in Google Scholar
[6] L. Guth, Lipschitz maps from surfaces, Geom. Funct. Anal. 15 (2005), no. 5, 1052–1090. 10.1007/s00039-005-0532-9Suche in Google Scholar
[7] L. Guth, Volumes of balls in Riemannian manifolds and Uryson width, J. Topol. Anal. 9 (2017), no. 2, 195–219. 10.1142/S1793525317500029Suche in Google Scholar
[8]
H. Lebesgue,
Sur la non-applicabilité de deux domaines appartenant respectivement à des espaces à n et
[9] Y. Liokumovich, B. Lishak, A. Nabutovsky and R. Rotman, Filling metric spaces, preprint (2019), https://arxiv.org/abs/1905.06522. 10.1215/00127094-2021-0039Suche in Google Scholar
[10] A. Nabutovsky, Linear bounds for constants in Gromov’s systolic inequality and related results, preprint (2019), https://arxiv.org/abs/1909.12225. Suche in Google Scholar
[11] P. Papasoglu, Uryson width and volume, Geom. Funct. Anal. 30 (2020), no. 2, 574–587. 10.1007/s00039-020-00533-5Suche in Google Scholar
[12] J. Y. Shi, The Kazhdan–Lusztig cells in certain affine Weyl groups, Lecture Notes in Math. 1179, Springer, Berlin 1986. 10.1007/BFb0074968Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces
- A support theorem for\break the Hitchin fibration: The case of GLn and KC
- Singular tuples of matrices is not a null cone (and the symmetries of algebraic varieties)
- Quasiexcellence implies strong generation
- Serre–Tate theory for Calabi–Yau varieties
- Vanishing and estimation results for Hodge numbers
- Torus actions, maximality, and non-negative curvature
- Local-to-global Urysohn width estimates
Artikel in diesem Heft
- Frontmatter
- Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces
- A support theorem for\break the Hitchin fibration: The case of GLn and KC
- Singular tuples of matrices is not a null cone (and the symmetries of algebraic varieties)
- Quasiexcellence implies strong generation
- Serre–Tate theory for Calabi–Yau varieties
- Vanishing and estimation results for Hodge numbers
- Torus actions, maximality, and non-negative curvature
- Local-to-global Urysohn width estimates
