Lipschitz Carnot-Carathéodory Structures and their Limits
Antonelli, G., Le Donne, E., & Nicolussi Golo, S. (2023). Lipschitz Carnot-Carathéodory Structures and their Limits. Journal of Dynamical and Control Systems, 29, 805-854. https://doi.org/10.1007/s10883-022-09613-1
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Journal of Dynamical and Control SystemsDate
2023Discipline
Geometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköGeometric Analysis and Mathematical PhysicsMathematicsAnalysis and Dynamics Research (Centre of Excellence)Copyright
© The Author(s) 2022
In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.
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Springer Science and Business Media LLCISSN Search the Publication Forum
1079-2724Keywords
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https://converis.jyu.fi/converis/portal/detail/Publication/164991042
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Related funder(s)
European Commission; Research Council of FinlandFunding program(s)
ERC Starting Grant; Academy Research Fellow, AoF; Academy Project, AoF; Research costs of Academy Research Fellow, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about funding
Open access funding provided by Scuola Normale Superiore within the CRUI-CARE Agreement. The authors are partially supported by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). E.L.D. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’). S.N.G. has been supported by the Academy of Finland (grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’, 328846 ‘Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups’, and grant 314172 ‘Quantitative rectifiability in Euclidean and non-Euclidean spaces’). ...License
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