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Lipschitz Carnot-Carathéodory Structures and their Limits.

Gioacchino Antonelli1, Enrico Le Donne2,3, Sebastiano Nicolussi Golo3

  • 1Courant Institute of Mathematical Sciences, NYU, 251 Mercer Street, New York, NY 10012 USA.

Journal of Dynamical and Control Systems
|September 25, 2023
PubMed
Summary
This summary is machine-generated.

This study proves that distances from converging Lipschitz vector fields and norms converge to the Carnot-Carathéodory distance under mild conditions. Convergence is uniform when the limit distance is boundedly compact.

Keywords:
Lipschitz vector fieldsMitchell’s TheoremSub-Finsler geometrySub-Riemannian geometry

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Area of Science:

  • Differential Geometry
  • Analysis on Metric Spaces
  • Geometric Measure Theory

Background:

  • Lipschitz vector fields and continuously varying norms on smooth manifolds are fundamental in geometric analysis.
  • Understanding the convergence of associated distances is crucial for developing robust geometric tools.

Purpose of the Study:

  • To analyze the convergence of distances derived from converging structures of Lipschitz vector fields and norms.
  • To establish conditions under which these distances converge to the limit Carnot-Carathéodory distance.

Main Methods:

  • Utilizing uniform convergence on compact subsets for vector fields and norms.
  • Applying controllability assumptions on the limit vector-field structure.
  • Investigating the impact of the limit distance's compactness properties on convergence uniformity.

Main Results:

  • Demonstrated local uniform convergence of distances to the Carnot-Carathéodory distance under mild controllability.
  • Proved uniform convergence on compact sets when the limit distance is boundedly compact.
  • Provided a counterexample for non-uniform convergence when the limit distance is not boundedly compact.

Conclusions:

  • The convergence of distances is well-behaved under specific uniformity and controllability conditions.
  • The study extends convergence results to sub-Finsler geometry and Lie groups, with applications in geometric analysis.