Subdifferentials and minimizing Sard conjecture in sub-Riemannian geometry

Rifford, Ludovic

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1195-1244

We use techniques from nonsmooth analysis and geometric measure theory to provide new examples of complete sub-Riemannian structures satisfying the Minimizing Sard conjecture. In particular, we show that complete sub-Riemannian structures associated with distributions of co-rank $2$ or generic distributions of rank $\ge 2$ satisfy the Minimizing Sard conjecture.

On utilise des techniques d’analyse non-lisse et de théorie géométrique de la mesure pour produire de nouveaux exemples de structures sous-riemanniennes complètes vérifiant la conjecture de Sard minimisante. On démontre en particulier que les structures sous-riemanniennes complètes associées à des distributions de co-rang $2$ ou génériques de rang $\ge 2$ vérifient la conjecture de Sard minimisante.

Classification : 53C17, 49J52
Keywords: Sub-Riemannian geometry, nonsmooth analysis, geometric measure theory
Mots-clés : Géométrie sous-riemannienne, analyse non-lisse, théorie géométrique de la mesure

[1] Agrachev, Andrei Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Sem. Mat. Univ. e Politec. Torino, Volume 56 (1998) no. 4, p. 1-12 (2001) Control theory and its applications (Grado, 1998) | Zbl | MR

[2] Agrachev, Andrei Any sub-Riemannian metric has points of smoothness, Dokl. Akad. Nauk, Volume 424 (2009) no. 3, pp. 295-298 | DOI

[3] Agrachev, Andrei Some open problems, Geometric control theory and sub-Riemannian geometry (Springer INdAM Ser.), Volume 5, Springer, Cham, 2014, pp. 1-13 | Zbl | MR | DOI

[4] Agrachev, Andrei; Barilari, Davide; Boscain, Ugo A comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint, Cambridge Studies in Advanced Math., 181, Cambridge University Press, Cambridge, 2020

[5] Agrachev, Andrei; Gauthier, Jean-Paul On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré C Anal. Non Linéaire, Volume 18 (2001) no. 3, pp. 359-382 | Zbl | Numdam | MR | DOI

[6] Agrachev, Andrei; Lee, Paul Optimal transportation under nonholonomic constraints, Trans. Amer. Math. Soc., Volume 361 (2009) no. 11, pp. 6019-6047 | Zbl | MR | DOI

[7] Agrachev, Andrei; Lee, Paul W. Y. Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds, Math. Ann., Volume 360 (2014) no. 1-2, pp. 209-253 | Zbl | MR | DOI

[8] Agrachev, Andrei; Sachkov, Yuri L. Control theory from the geometric viewpoint. Control Theory and Optimization, II, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004 | DOI

[9] Agrachev, Andrei; Sarychev, Andrei V. Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré C Anal. Non Linéaire, Volume 13 (1996) no. 6, pp. 635-690 | Zbl | Numdam | MR | DOI

[10] Agrachev, Andrei; Sarychev, Andrei V. Sub-Riemannian metrics: minimality of abnormal geodesics versus subanalyticity, ESAIM Control Optim. Calc. Var., Volume 4 (1999), pp. 377-403 | Zbl | MR | DOI

[11] Badreddine, Zeinab; Rifford, Ludovic Measure contraction properties for two-step analytic sub-Riemannian structures and Lipschitz Carnot groups, Ann. Inst. Fourier (Grenoble), Volume 70 (2020) no. 6, pp. 2303-2330 http://aif.cedram.org/... | Zbl | Numdam | MR | DOI

[12] Barilari, Davide; Rizzi, Luca Sharp measure contraction property for generalized H-type Carnot groups, Commun. Contemp. Math., Volume 20 (2018) no. 6, 1750081, 24 pages | Zbl | MR | DOI

[13] Bellaïche, A. The tangent space in sub-Riemannian geometry, J. Math. Sci. (New York), Volume 83 (1997) no. 4, pp. 461-476 (Dynamical systems, 3) | Zbl | MR | DOI

[14] Belotto da Silva, André; Figalli, Alessio; Parusiński, Adam; Rifford, Ludovic Strong Sard conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3, Invent. Math., Volume 229 (2022) no. 1, pp. 395-448 | Zbl | MR | DOI

[15] Belotto da Silva, André; Parusiński, Adam; Rifford, Ludovic Abnormal subanalytic distributions and minimal rank Sard conjecture, 2022 | arXiv

[16] Belotto da Silva, André; Rifford, Ludovic The Sard conjecture on Martinet surfaces, Duke Math. J., Volume 167 (2018) no. 8, pp. 1433-1471 | Zbl | MR | DOI

[17] Boarotto, Francesco; Nalon, Luca; Vittone, Davide The Sard problem in step $2$ and in filiform Carnot groups, 2022 | arXiv

[18] Boarotto, Francesco; Vittone, Davide A dynamical approach to the Sard problem in Carnot groups, J. Differential Equations, Volume 269 (2020) no. 6, pp. 4998-5033 | Zbl | MR | DOI

[19] Chitour, Y.; Jean, F.; Trélat, E. Genericity results for singular curves, J. Differential Geom., Volume 73 (2006) no. 1, pp. 45-73 http://projecteuclid.org/euclid.jdg/1146680512 | Zbl | MR

[20] Clarke, Francis H. Functional analysis, calculus of variations and optimal control, Graduate Texts in Math., 264, Springer, London, 2013 | DOI

[21] Clarke, Francis H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, P. R. Nonsmooth analysis and control theory, Graduate Texts in Math., 178, Springer-Verlag, New York, 1998

[22] Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions, Textbooks in Math., CRC Press, Boca Raton, FL, 2015 | DOI

[23] Federer, Herbert Geometric measure theory, Grundlehren Math. Wissen., 153, Springer-Verlag New York, Inc., New York, 1969

[24] Góra, P.; Stern, R. J. Subdifferential analysis of the Van der Waerden function, J. Convex Anal., Volume 18 (2011) no. 3, pp. 699-705 | Zbl | MR

[25] Hamilton, Richard S. The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), Volume 7 (1982) no. 1, pp. 65-222 | Zbl | MR | DOI

[26] Hanson, E. H. A new proof of a theorem of Denjoy, Young, and Saks, Bull. Amer. Math. Soc., Volume 40 (1934) no. 10, pp. 691-694 | Zbl | MR | DOI

[27] Juillet, Nicolas Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Internat. Math. Res. Notices (2009) no. 13, pp. 2347-2373 | Zbl | MR | DOI

[28] Juillet, Nicolas Sub-Riemannian structures do not satisfy Riemannian Brunn-Minkowski inequalities, Rev. Mat. Iberoamericana, Volume 37 (2021) no. 1, pp. 177-188 | Zbl | MR | DOI

[29] Le Donne, Enrico; Montgomery, Richard; Ottazzi, Alessandro; Pansu, Pierre; Vittone, Davide Sard property for the endpoint map on some Carnot groups, Ann. Inst. H. Poincaré C Anal. Non Linéaire, Volume 33 (2016) no. 6, pp. 1639-1666 | Zbl | Numdam | MR | DOI

[30] Lee, Paul W. Y. On measure contraction property without Ricci curvature lower bound, Potential Anal., Volume 44 (2016) no. 1, pp. 27-41 | Zbl | MR | DOI

[31] Lee, Paul W. Y.; Li, Chengbo; Zelenko, Igor Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds, Discrete Contin. Dynam. Systems, Volume 36 (2016) no. 1, pp. 303-321 | Zbl | MR | DOI

[32] Montgomery, Richard A tour of subriemannian geometries, their geodesics and applications, Math. Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002 | DOI

[33] Ohta, Shin-ichi On the measure contraction property of metric measure spaces, Comment. Math. Helv., Volume 82 (2007) no. 4, pp. 805-828 | Zbl | MR | DOI

[34] Ottazzi, Alessandro; Vittone, Davide On the codimension of the abnormal set in step two Carnot groups, ESAIM Control Optim. Calc. Var., Volume 25 (2019), 18, 17 pages | Zbl | MR | DOI

[35] Rifford, Ludovic Ricci curvatures in Carnot groups, Math. Control Relat. Fields, Volume 3 (2013) no. 4, pp. 467-487 | Zbl | MR | DOI

[36] Rifford, Ludovic Sub-Riemannian geometry and optimal transport, SpringerBriefs in Math., Springer, Cham, 2014 | DOI

[37] Rifford, Ludovic Singulières minimisantes en géométrie sous-Riemannienne, Séminaire Bourbaki 2015/2016 (Astérisque), Volume 390, Société Mathématique de France, Paris, 2017, pp. 277-301 (Exp. No. 1113) | Zbl

[38] Rifford, Ludovic; Trélat, E. Morse-Sard type results in sub-Riemannian geometry, Math. Ann., Volume 332 (2005) no. 1, pp. 145-159 | Zbl | MR | DOI

[39] Rizzi, Luca Measure contraction properties of Carnot groups, Calc. Var. Partial Differential Equations, Volume 55 (2016) no. 3, 60, 20 pages | Zbl | MR | DOI

[40] Rockafellar, R. Tyrrell; Wets, Roger J.-B. Variational analysis, Grundlehren Math. Wissen., 317, Springer-Verlag, Berlin, 1998 | DOI

[41] Sakai, Takashi Riemannian geometry, Translations of Math.Monographs, 149, American Mathematical Society, Providence, RI, 1996 | DOI

[42] Subbotin, Andreĭ I. Generalized solutions of first-order PDEs. The dynamical optimization perspective, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995 | DOI

[43] Trélat, E. Some properties of the value function and its level sets for affine control systems with quadratic cost, J. Dynam. Control Systems, Volume 6 (2000) no. 4, pp. 511-541 | Zbl | MR | DOI