Inventiones mathematicae

, Volume 215, Issue 3, pp 977–1038 | Cite as

Sub-Riemannian interpolation inequalities

  • Davide BarilariEmail author
  • Luca Rizzi


We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients, whose properties are remarkably different with respect to the Riemannian case. As a byproduct, we characterize the cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex, answering to a question raised by Figalli and Rifford (Geom Funct Anal 20(1):124–159, 2010). As an application, we deduce sharp and intrinsic Borell–Brascamp–Lieb and geodesic Brunn–Minkowski inequalities in the aforementioned setting. For the case of the Heisenberg group, we recover in an intrinsic way the results recently obtained by Balogh et al. (Calc Var Part Differ Equ 57(2):61, 2018), and we extend them to the class of generalized H-type Carnot groups. Our results do not require the distribution to have constant rank, yielding for the particular case of the Grushin plane a sharp measure contraction property and a sharp Brunn–Minkowski inequality.

Mathematics Subject Classification

53C17 49J15 49Q20 



This work was supported by the Grant ANR-15-CE40-0018 of the ANR, and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with the “FMJH Program Gaspard Monge in optimization and operation research”. This work has been supported by the ANR project ANR-15-IDEX-02.

We wish to thank Andrei Agrachev, Luigi Ambrosio, Zoltan Balogh, Ludovic Rifford and Kinga Sipos for stimulating discussions. We also thank the anonymous referees for their comments.


  1. 1.
    Agrachev, A., Barilari, D., Boscain, U.: On the Hausdorff volume in sub-Riemannian geometry. Calc. Var. Part. Differ. Equ. 43(3–4), 355–388 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agrachev, A., Barilari, D., Boscain, U.: Introduction to geodesics in sub-Riemannian geometry. Geom. Anal. Dyn. Sub-Riemannian Manifolds 2, 1–83 (2016). (English)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Agrachev, A., Barilari, D., Boscain, U.: A Comprehensive Introduction to Sub-Riemannian Geometry. Universiy Press, Cambridge (2019)zbMATHGoogle Scholar
  4. 4.
    Agrachev, A., Bonnard, B., Chyba, M., Kupka, I.: Sub-Riemannian sphere in Martinet flat case. ESAIM Control Optim. Calc. Var. 2, 377–448 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Agrachev, A., Barilari, D., Rizzi, L.: Curvature: a variational approach. Mem. AMS 256, 1225 (2018)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Agrachev, A., Boscain, U., Sigalotti, M.: A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discret. Cont. Dyn. Syst. 20(4), 801–822 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Agrachev, A.: Any sub-Riemannian metric has points of smoothness. Dokl. Akad. Nauk 424(3), 295–298 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)zbMATHGoogle Scholar
  9. 9.
    Agrachev, A., Lee, P.W.Y.: Optimal transportation under nonholonomic constraints. Trans. Am. Math. Soc. 361(11), 6019–6047 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Agrachev, A., Lee, P.W.Y.: Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds. Math. Ann. 360(1–2), 209–253 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ambrosio, L., Rigot, S.: Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208(2), 261–301 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Agrachev, A., Zelenko, I.: Geometry of Jacobi curves. I. J. Dyn. Control Syst. 8(1), 93–140 (2002)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Agrachev, A., Zelenko, I.: Geometry of Jacobi curves. II. J. Dyn. Control Syst. 8(2), 167–215 (2002)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Baudoin, F., Garofalo, N.: Curvature-dimension inequalities and RICCI lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc. (JEMS) 19(1), 151–219 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Balogh, Z.M., Kristály, A., Sipos, K.: Jacobian determinant inequality on Corank 1 Carnot groups with applications (2017) ArXiv e-prints arXiv:1701.08831
  16. 16.
    Balogh, Z.M., Kristály, A., Sipos, K.: Geometric inequalities on Heisenberg groups. Calc. Var. Part. Differ. Equ. 57(2), 61 (2018)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Baudoin, F., Kim, B., Wang, J.: Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves. Commun. Anal. Geom. 24(5), 913–937 (2016)zbMATHGoogle Scholar
  18. 18.
    Beschastnyi, I., Medvedev, A.: Left-invariant sub-Riemannian engel structures: abnormal geodesics and integrability. SIAM J. Control Optim. 56(5), 3524–3537 (2018)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Barilari, D., Rizzi, L.: Comparison theorems for conjugate points in sub-Riemannian geometry. ESAIM Control Optim. Calc. Var. 22(2), 439–472 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Badreddine, Z., Rifford, L.: Measure contraction properties for two-step sub-Riemannian structures and medium-fat Carnot groups (2017). ArXiv e-prints arXiv:1712.09900
  21. 21.
    Barilari, D., Rizzi, L.: On Jacobi fields and a canonical connection in sub-Riemannian geometry. Arch. Math. 53(2), 77–92 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Barilari, D., Rizzi, L.: Sharp measure contraction property for generalized H-type Carnot groups. Commun. Contemp. Math. 20(6), 1750081 (2018)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52(4), 411–452 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Chitour, Y., Jean, F., Trélat, E.: Genericity results for singular curves. J. Differ. Geom. 73(1), 45–73 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Cannarsa, P., Rifford, L.: Semiconcavity results for optimal control problems admitting no singular minimizing controls. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(4), 773–802 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser Boston Inc, Boston (2004)zbMATHGoogle Scholar
  28. 28.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised, Textbooks in Mathematics. CRC Press, Boca Raton, FL (2015)zbMATHGoogle Scholar
  29. 29.
    Figalli, A., Juillet, N.: Absolute continuity of Wasserstein geodesics in the Heisenberg group. J. Funct. Anal. 255(1), 133–141 (2008)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Figalli, A., Rifford, L.: Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20(1), 124–159 (2010)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gardner, R.J.: The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39(3), 355–405 (2002)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Jean, F.: Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning. SpringerBriefs in Mathematics. Springer, Cham (2014)zbMATHGoogle Scholar
  33. 33.
    Juillet, N.: Geometric inequalities and generalized Ricci bounds in the Heisenberg group. Int. Math. Res. Not. IMRN 13, 2347–2373 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 editionzbMATHGoogle Scholar
  35. 35.
    Lee, P.W.Y.: Displacement interpolations from a Hamiltonian point of view. J. Funct. Anal. 265(12), 3163–3203 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lee, P.W.Y., Li, C., Zelenko, I.: Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discret. Cont. Dyn. Syst. 36(1), 303–321 (2016)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Leonardi, G.P., Masnou, S.: On the isoperimetric problem in the Heisenberg group \(\mathbb{H}^n\). Ann. Mat. Pura Appl. (4) 184(4), 533–553 (2005)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169(3), 903–991 (2009)MathSciNetzbMATHGoogle Scholar
  39. 39.
    McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Mitchell, J.: On Carnot-Carathéodory metrics. J. Differential Geom. 21(1), 35–45 (1985)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Montanari, A., Morbidelli, D.: On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups. J. Math. Anal. Appl. 444(2), 1652–1674 (2016)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Montanari, A., Morbidelli, D.: On the subRiemannian cut locus in a model of free two-step Carnot group. Calc. Var. Part. Differ. Equ. 56(2), 36 (2017)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Dover Publications, Inc., New York (1992). Reprint of the 1969 editionzbMATHGoogle Scholar
  44. 44.
    Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI (2002)Google Scholar
  45. 45.
    Monti, R.: Brunn–Minkowski and isoperimetric inequality in the Heisenberg group. Ann. Acad. Sci. Fenn. Math. 28(1), 99–109 (2003)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Munive, I.H.: Sub-Riemannian curvature of Carnot groups with rank-two distributions. J. Dyn. Control Syst. 23(4), 779–814 (2017)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Ohta, S.: On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82(4), 805–828 (2007)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Ohta, S.: Finsler interpolation inequalities. Calc. Var. Part. Differ. Equ. 36(2), 211–249 (2009)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Rifford, L.: Ricci curvatures in Carnot groups. Math. Control Relat. Fields 3(4), 467–487 (2013)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Rifford, L.: Sub-Riemannian Geometry and Optimal Transport. Springer Briefs in Mathematics. Springer, Cham (2014)zbMATHGoogle Scholar
  51. 51.
    Rizzi, L.: Measure contraction properties of Carnot groups. Calc. Var. Part. Differ. Equ. 55(3), 60 (2016)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Rizzi, L.: A counterexample to gluing theorems for mcp metric measure spaces. Bull. Lond. Math. Soc. 50(5), 781–790 (2018)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Rizzi, L., Serres, U.: On the cut locus of free, step two Carnot groups. Proc. Am. Math. Soc. 145(12), 5341–5357 (2017)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Rifford, L., Trélat, E.: Morse-Sard type results in sub-Riemannian geometry. Math. Ann. 332(1), 145–159 (2005)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Ritoré, M., Yepes Nicolás, J.: Brunn–Minkowski inequalities in product metric measure spaces. Adv. Math. 325, 824–863 (2018)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Sarychev, A.V.: Index of second variation of a control system. Mat. Sb. (N.S.) 155(3), 464–486 (1980)MathSciNetGoogle Scholar
  57. 57.
    Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Villani, C.: Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer, Berlin (2009). Old and newGoogle Scholar
  60. 60.
    Villani, C.: Séminaire Bourbaki, Volume 2016/2017, exposé 1127, Astérisque (2017)Google Scholar
  61. 61.
    Zelenko, I., Li, C.: Differential geometry of curves in Lagrange Grassmannians with given Young diagram. Differ. Geom. Appl. 27(6), 723–742 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR CNRS 7586Université Paris DiderotParis Cedex 13France
  2. 2.Univ. Grenoble Alpes, CNRS, IFGrenobleFrance

Personalised recommendations