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Existence of tangent lines to Carnot–Carathéodory geodesics

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Abstract

We show that length minimizing curves in Carnot–Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line.

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Acknowledgements

We are grateful to the anonymous referee for carefully reading the paper and for helpful advice. We thank L. Ambrosio for several discussions and for being an invaluable mentor and friend.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121, Padua, Italy

    Roberto Monti & Davide Vittone

  2. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Alessandro Pigati

  3. Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Alessandro Pigati

Authors
  1. Roberto Monti
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  2. Alessandro Pigati
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  3. Davide Vittone
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Corresponding author

Correspondence to Davide Vittone.

Additional information

Communicated by L. Ambrosio.

R. M. and D. V. are supported by MIUR (Italy) and University of Padova STARS Project “Sub-Riemannian Geometry and Geometric Measure Theory Issues: Old and New” (SUGGESTION). D. V. is also supported by University of Padova Project Networking and INdAM-GNAMPA Project 2017 “Campi vettoriali, superfici e perimetri in geometrie singolari”.

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Monti, R., Pigati, A. & Vittone, D. Existence of tangent lines to Carnot–Carathéodory geodesics. Calc. Var. 57, 75 (2018). https://doi.org/10.1007/s00526-018-1361-7

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