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Non-minimality of corners in subriemannian geometry

Abstract

We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev’s list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics.

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Notes

  1. Alternatively, one can use the formula [26, page 114]

    $$\begin{aligned} \mathrm {C}_{\exp \left( X\right) }\left( \exp (Y)\right) = \exp (\mathrm {Ad}_{\exp (X)}Y) = \exp (e^{\mathrm {ad}_X}Y). \end{aligned}$$
  2. We remark that for rank-varying distributions, desingularizations of curves with corner-type singularities need not have one-sided derivatives.

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Acknowledgments

The authors thank A. Ottazzi, D. Vittone, and the anonymous referees for their helpful remarks. E.L.D. acknowledges the support of the Academy of Finland project no. 288501.

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Correspondence to Enrico Le Donne.

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Hakavuori, E., Le Donne, E. Non-minimality of corners in subriemannian geometry. Invent. math. 206, 693–704 (2016). https://doi.org/10.1007/s00222-016-0661-9

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  • DOI: https://doi.org/10.1007/s00222-016-0661-9

Mathematics Subject Classification

  • 53C17
  • 49K21
  • 28A75