A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of... References[1] Andrei Agrachev and Davide Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst. 18 (2012), no. 1, 21-44.[2] Andrei Agrachev, Davide Barilari, and Ugo Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 355-388.[3] , Introduction to Riemannian and Sub-Riemannian geometry, Manuscript (2015).[4] Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1-80.[5] , Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527-555.[6] Luigi Ambrosio, Bruce Kleiner, and Enrico Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal. 19 (2009), no. 3, 509-540.10.1007/s12220-009-9068-9[7] Luigi Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 (2001), no. 1, 51-67.[8] , Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal. 10 (2002), no. 2-3, 111-128, Calculus of variations, nonsmooth analysis and related topics.[9] Luigi Ambrosio, Francesco Serra Cassano, and Davide Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal. 16 (2006), no. 2, 187-232.10.1007/BF02922114[10] Dmitri˘ı Burago, Yuri˘ı Burago, and Sergei˘ı Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.[11] André Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1-78.[12] Costante Bellettini and Enrico Le Donne, Regularity of sets with constant horizontal normal in the Engel group, Comm. Anal. Geom. 21 (2013), no. 3, 469-507.10.4310/CAG.2013.v21.n3.a1[13] Valeri˘ı N. Berestovski˘ı, Homogeneous manifolds with an intrinsic metric. I, Sibirsk. Mat. Zh. 29 (1988), no. 6, 17-29.[14] Yuri˘ı Burago, Mikhail Gromov, and Grigori˘ı Perel0man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3-51, 222.[15] Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127-183.[16] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.[17] Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469-481.10.1090/S0894-0347-1991-1096169-1[18] Marc Bourdon and Hervé Pajot, Rigidity of quasi-isometries for somehyperbolic buildings, Comment.Math. Helv. 75 (2000), no. 4, 701-736.10.1007/s000140050146[19] Davide Barilari and Luca Rizzi, A formula for Popp’s volume in sub-Riemannian geometry, Anal. Geom. Metr. Spaces 1 (2013), 42-57.[20] Emmanuel Breuillard and Enrico Le Donne, Nilpotent groups, asymptotic cones and subFinsler geometry, In preparation.[21] , On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19220-19226.[22] Vittorio Barone Adesi, Francesco Serra Cassano, and Davide Vittone, The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations, Calc. Var. Partial Differential Equations 30 (2007), no. 1, 17-49.[23] C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann. 67 (1909), no. 3, 355-386.[24] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406-480.[25] Luca Capogna and Michael Cowling, Conformality and Q-harmonicity in Carnot groups, Duke Math. J. 135 (2006), no. 3, 455-479.[26] Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007.[27] Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990, Basic theory and examples.[28] Jeff Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428-517.10.1007/s000390050094[29] Jeff Cheeger and Bruce Kleiner, On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces, Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 129-152.10.1142/9789812772688_0006[30] , Differentiating maps into L1, and the geometry of BV functions, Ann. of Math. (2) 171 (2010), no. 2, 1347-1385.[31] , Metric differentiation, monotonicity and maps to L1, Invent. Math. 182 (2010), no. 2, 335-370.[32] Luca Capogna and Enrico Le Donne, Smoothness of subRiemannian isometries, Amer. J.Math. 138 (2016), no. 5, 1439-1454.[33] Daniel R. Cole and Scott D. Pauls, C1 hypersurfaces of the Heisenberg group are N-rectifiable, Houston J. Math. 32 (2006), no. 3, 713-724 (electronic). MR MR2247905 (2007f:53032)[34] D. Danielli, N. Garofalo, and D. M. Nhieu, A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing, Amer. J. Math. 130 (2008), no. 2, 317-339.[35] Katrin Fässler, Pekka Koskela, and Enrico Le Donne, Nonexistence of quasiconformal maps between certain metric measure spaces, Int. Math. Res. Not. IMRN (2015), no. 16, 6968-6987.[36] G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups,Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J., 1982.[37] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003), no. 3, 421-466.[38] Roberta Ghezzi and Frédéric Jean, Hausdorff measure and dimensions in non equiregular sub-Riemannian manifolds, Geometriccontrol theory and sub-Riemannian geometry 5 (2014), 201-218.[39] Andrew M. Gleason, Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193-212.[40] Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981), no. 53, 53-73.[41] Mikhail Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79-323.[42] , Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, BirkhäuserBoston Inc., Boston, MA, 1999, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.[43] Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groupsof rank one, Inst. Hautes Études Sci. Publ. Math. (1992), no. 76, 165-246.[44] Ursula Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics, J. Differential Geom. 32 (1990), no. 3, 819-850.[Hei74] Ernst Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34. MR 0353210[Hei95] Juha Heinonen, Calculus on Carnot groups, Fall School in Analysis (Jyväskylä, 1994), Report, vol. 68, Univ. Jyväskylä, Jyväskylä, 1995, pp. 1-31.[47] , Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001.[48] Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, ActaMath. 181 (1998), no. 1, 1-61.[49] Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101.[50] Waldemar Hebisch and Adam Sikora, A smooth subadditive homogeneous norm on a homogeneous group, Studia Math.96 (1990), no. 3, 231-236.[51] Frédéric Jean, Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, Springer Briefs inMathematics, Springer, Cham, 2014.[52] David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503-523.[53] Ilya Kapovich and Nadia Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 39-93.[54] Bruce Kleiner and Bernhard Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. HautesÉtudes Sci. Publ. Math. (1997), no. 86, 115-197 (1998).[55] Ville Kivioja and Enrico Le Donne, Isometries of nilpotent metric groups, J. Éc. polytech. Math. 4 (2017), 473-482. MR3646026[56] A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), no. 2, 309-338.[57] , Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), no. 1, 1-87.[58] Bernd Kirchheim and Francesco Serra Cassano, Rectifiability and parameterization of intrinsic regular surfaces in theHeisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 871-896.[59] Tomi J. Laakso, Plane with A1-weighted metric not bi-Lipschitz embeddable to RN, Bull. London Math. Soc. 34 (2002), no. 6, 667-676.[60] Enrico Le Donne, Lipschitz and path isometric embeddings of metric spaces, Geom. Dedicata 166 (2013), 47-66.[61] , Lecture notes on sub-Riemannian geometry, Manuscript (2015).[62] , A metric characterization of Carnot groups, Proc. Amer. Math. Soc. 143 (2015), no. 2, 845-849.[63] Enrico Le Donne and Sebastiano Nicolussi Golo, Regularity properties of spheres in homogeneous groups, Trans. Amer. Math. Soc. (2016).[64] , Homogeneous distances on Lie groups, Preprint, https://sites.google.com/site/enricoledonne/ (2017).[65] Enrico Le Donne and Alessandro Ottazzi, Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds, J. Geom. Anal. 26 (2016), no. 1, 330-345.[66] Urs Lang and Conrad Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. 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MR 0058607 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Geometry in Metric Spaces de Gruyter

A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

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References[1] Andrei Agrachev and Davide Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst. 18 (2012), no. 1, 21-44.[2] Andrei Agrachev, Davide Barilari, and Ugo Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 355-388.[3] , Introduction to Riemannian and Sub-Riemannian geometry, Manuscript (2015).[4] Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1-80.[5] , Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527-555.[6] Luigi Ambrosio, Bruce Kleiner, and Enrico Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal. 19 (2009), no. 3, 509-540.10.1007/s12220-009-9068-9[7] Luigi Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 (2001), no. 1, 51-67.[8] , Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal. 10 (2002), no. 2-3, 111-128, Calculus of variations, nonsmooth analysis and related topics.[9] Luigi Ambrosio, Francesco Serra Cassano, and Davide Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal. 16 (2006), no. 2, 187-232.10.1007/BF02922114[10] Dmitri˘ı Burago, Yuri˘ı Burago, and Sergei˘ı Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.[11] André Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1-78.[12] Costante Bellettini and Enrico Le Donne, Regularity of sets with constant horizontal normal in the Engel group, Comm. Anal. Geom. 21 (2013), no. 3, 469-507.10.4310/CAG.2013.v21.n3.a1[13] Valeri˘ı N. Berestovski˘ı, Homogeneous manifolds with an intrinsic metric. I, Sibirsk. Mat. Zh. 29 (1988), no. 6, 17-29.[14] Yuri˘ı Burago, Mikhail Gromov, and Grigori˘ı Perel0man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3-51, 222.[15] Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127-183.[16] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.[17] Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469-481.10.1090/S0894-0347-1991-1096169-1[18] Marc Bourdon and Hervé Pajot, Rigidity of quasi-isometries for somehyperbolic buildings, Comment.Math. Helv. 75 (2000), no. 4, 701-736.10.1007/s000140050146[19] Davide Barilari and Luca Rizzi, A formula for Popp’s volume in sub-Riemannian geometry, Anal. Geom. Metr. Spaces 1 (2013), 42-57.[20] Emmanuel Breuillard and Enrico Le Donne, Nilpotent groups, asymptotic cones and subFinsler geometry, In preparation.[21] , On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19220-19226.[22] Vittorio Barone Adesi, Francesco Serra Cassano, and Davide Vittone, The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations, Calc. Var. Partial Differential Equations 30 (2007), no. 1, 17-49.[23] C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann. 67 (1909), no. 3, 355-386.[24] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406-480.[25] Luca Capogna and Michael Cowling, Conformality and Q-harmonicity in Carnot groups, Duke Math. J. 135 (2006), no. 3, 455-479.[26] Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007.[27] Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990, Basic theory and examples.[28] Jeff Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428-517.10.1007/s000390050094[29] Jeff Cheeger and Bruce Kleiner, On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces, Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 129-152.10.1142/9789812772688_0006[30] , Differentiating maps into L1, and the geometry of BV functions, Ann. of Math. (2) 171 (2010), no. 2, 1347-1385.[31] , Metric differentiation, monotonicity and maps to L1, Invent. Math. 182 (2010), no. 2, 335-370.[32] Luca Capogna and Enrico Le Donne, Smoothness of subRiemannian isometries, Amer. J.Math. 138 (2016), no. 5, 1439-1454.[33] Daniel R. Cole and Scott D. Pauls, C1 hypersurfaces of the Heisenberg group are N-rectifiable, Houston J. Math. 32 (2006), no. 3, 713-724 (electronic). MR MR2247905 (2007f:53032)[34] D. Danielli, N. Garofalo, and D. M. Nhieu, A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing, Amer. J. Math. 130 (2008), no. 2, 317-339.[35] Katrin Fässler, Pekka Koskela, and Enrico Le Donne, Nonexistence of quasiconformal maps between certain metric measure spaces, Int. Math. Res. Not. IMRN (2015), no. 16, 6968-6987.[36] G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups,Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J., 1982.[37] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003), no. 3, 421-466.[38] Roberta Ghezzi and Frédéric Jean, Hausdorff measure and dimensions in non equiregular sub-Riemannian manifolds, Geometriccontrol theory and sub-Riemannian geometry 5 (2014), 201-218.[39] Andrew M. Gleason, Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193-212.[40] Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981), no. 53, 53-73.[41] Mikhail Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79-323.[42] , Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, BirkhäuserBoston Inc., Boston, MA, 1999, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.[43] Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groupsof rank one, Inst. Hautes Études Sci. Publ. Math. (1992), no. 76, 165-246.[44] Ursula Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics, J. Differential Geom. 32 (1990), no. 3, 819-850.[Hei74] Ernst Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34. MR 0353210[Hei95] Juha Heinonen, Calculus on Carnot groups, Fall School in Analysis (Jyväskylä, 1994), Report, vol. 68, Univ. 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