Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincaré inequality
Abstract
For every integer d≥ 3 and real number Q∈ (d,(d(d−1)+2)/2], we construct a Q-Ahlfors regular metric measure manifold (M,d,μ) of topological dimension d and analytic dimension $2$ that satisfies a Poincaré inequality. For every Q>(d(d−1)+2)/2, the construction yields a metric measure manifold (M,d,μ) that is doubling, satisfies a Poincaré inequality, and for which M has μ-essential Hausdorff dimension Q. The manifold is ℝd equipped with Lebesgue measure and a sub-Riemannian distance given by vector fields that are locally Hölder continuous but not Lipschitz continuous. These metrics give an uncountable family of new models that are genuinely distinct from the only known families of metric manifolds satisfying a Poincaré inequality: weighted and deformed Carnot groups and smooth sub-Riemannian spaces. The construction is modeled on a singular variant of a filiform Carnot group.
Related
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- A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds
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- Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
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BibTeX
@misc{erikssonbique2026nonsmooth,
title = {{Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincaré inequality}},
author = {Sylvester Eriksson-Bique and Enrico Le Donne},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17863},
eprint = {2610.17863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17863},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincaré inequality
AU - Sylvester Eriksson-Bique
AU - Enrico Le Donne
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17863
UR - https://arxiv.org/abs/2610.17863
AB - For every integer d≥ 3 and real number Q∈ (d,(d(d−1)+2)/2], we construct a Q-Ahlfors regular metric measure manifold (M,d,μ) of topological dimension d and analytic dimension $2$ that satisfies a Poincaré inequality. For every Q>(d(d−1)+2)/2, the construction yields a metric measure manifold (M,d,μ) that is doubling, satisfies a Poincaré inequality, and for which M has μ-essential Hausdorff dimension Q. The manifold is ℝ^d equipped with Lebesgue measure and a sub-Riemannian distance given by vector fields that are locally Hölder continuous but not Lipschitz continuous. These metrics give an uncountable family of new models that are genuinely distinct from the only known families of metric manifolds satisfying a Poincaré inequality: weighted and deformed Carnot groups and smooth sub-Riemannian spaces. The construction is modeled on a singular variant of a filiform Carnot group.
ER -