Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
| Autor: | Rizzi, Luca, Stefani, Giorgio |
|---|---|
| Přispěvatelé: | Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), ANR-18-CE40-0012,RAGE,Analyse Réelle et Géométrie(2018), European Project: 945655,GEOSUB, Rizzi, Luca, Analyse Réelle et Géométrie - - RAGE2018 - ANR-18-CE40-0012 - AAPG2018 - VALID, Geometric analysis of sub-Riemannian spaces through interpolation inequalities - GEOSUB - 945655 - INCOMING |
| Rok vydání: | 2023 |
| Předmět: |
Mathematics - Differential Geometry
Mathematics - Functional Analysis Mathematics - Metric Geometry Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] FOS: Mathematics Metric Geometry (math.MG) [MATH.MATH-MG] Mathematics [math]/Metric Geometry [math.MG] 53C17 54E45 28A75 [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG] Functional Analysis (math.FA) |
| DOI: | 10.48550/arxiv.2301.00735 |
| Popis: | We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-\'Emery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry-\'Emery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting. Comment: 25 pages |
| Databáze: | OpenAIRE |
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