Characterizations of intrinsic rectifiability in Heisenberg groups

Mattila, Pertti; Serapioni, Raul; Serra Cassano, Francesco

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 687-723

We define rectifiable sets in the Heisenberg groups as countable unions of Lipschitz images of subsets of a Euclidean space, in the case of low-dimensional sets, or as countable unions of subsets of intrinsic $C^1$ surfaces, in the case of low-codimensional sets. We characterize both low-dimensional rectifiable sets and low codimensional rectifiable sets with positive lower density, in terms of almost everywhere existence of approximate tangent subgroups or of tangent measures.

Classification : 28A75, 28C10

[1] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Oxford University Press, 2000. | MR | Zbl

[2] L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527–555. | MR | Zbl

[3] L. Ambrosio, B. Kleiner and E. Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal. 19 (2009), 509–540. | MR | Zbl

[4] L. Ambrosio, F. Serra Cassano and D. Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal. 16 (2006), 187–232. | MR | Zbl

[5] G. Arena and R. Serapioni, Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs, Calc. Var. Partial Differential Equations, 35 (2009), 17–49. | MR | Zbl

[6] Z. Balogh and K. S. Fässler, Rectifiability and Lipschitz extensions into the Heisenberg group, Math. Z. 263 (2009), 673–683. | MR | Zbl

[7] Z. Balogh, M. Rickly and F. Serra Cassano, Comparison of Hausdorff measures with respect to the Heisenberg metric, Publ. Mat. 47 (2003), 237–259. | MR | EuDML | Zbl

[8] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, “Stratified Lie Groups and Potential Theory for their Sub–Laplacians”, Springer Monographs in Mathematics, Springer-Verlag, 2007. | MR | Zbl

[9] L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, “An Introduction to the Heisenberg Group and the sub-Riemannian Isoperimetric Problem”, Birkhäuser, 2007. | MR | Zbl

[10] J. Cheeger and B. Kleiner, Differentiating maps into $L^1$ and the geometry of $BV$ functions, Ann. of Math. 171 (2010), 1347–1385. | MR | Zbl

[11] G. Citti and M. Manfredini, Implicit function theorem in Carnot Carathéodory spaces, Commun. Contemp. Math. 8, (2006), 253–293. | MR | Zbl

[12] H. Federer, “Geometric Measure Theory”, Springer-Verlag, 1969. | MR | Zbl

[13] G. B. Folland and E. M. Stein, “Hardy spaces in Homogeneous Groups”, Princeton University Press, 1982.

[14] B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479–531. | MR | Zbl

[15] B. Franchi, R. Serapioni and F. Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003), 421–466. | MR | Zbl

[16] B. Franchi, R. Serapioni and F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem, Comm. Anal. Geom. 11 (2003), 909–944. | MR | Zbl

[17] B. Franchi, R. Serapioni and F. Serra Cassano, Regular submanifolds, graphs and area formula in Heisenberg Groups, Adv. Math. 211 (2007), 152–203. | MR | Zbl

[18] B. Franchi, R. Serapioni and F. Serra Cassano, Differentiability of intrinsic Lipschitz functions within Heisenberg groups, J. Geom. Anal. DOI 10:1007/s12220-010-9178. | MR

[19] M. Gromov, Carnot-Carathéodory spaces seen from within, In: “Subriemannian Geometry”, Progress in Mathematics, Vol. 144, A. Bellaiche and J. Risler (eds.), Birkhäuser Verlag, Basel, 1996. | MR | Zbl

[20] B. Kirchheim and F. Serra Cassano, Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 871–896. | MR | EuDML | Zbl | Numdam

[21] V. Magnani, “Elements of Geometric Measure Theory on Sub-Riemannion groups”, Edizioni della Normale, Pisa, 2002. | MR

[22] V. Magnani, Contact equations, Lipschitz extensions and isoperimetric inequalities, preprint http://cvgmt.sns.it/cgi/get.cgi/papers/maga/ (2009). | MR

[23] P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces”, Cambridge University Press, 1995. | MR

[24] P. Mattila, Measures with unique tangent measures in metric groups, Math. Scand. 97 (2005), 298–308. | MR | Zbl

[25] J. Mitchell, On Carnot-Carathèodory metrics, J. Differential Geom. 21 (1985), 35–45. | MR | Zbl

[26] R. Montgomery, “A Tour of Sub-Riemannian Geometries, their Geodesics and Applications”, Mathematical Survey and Monographs, Vol. 91, American Mathematical Society, Providence RI, 2002. | MR

[27] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989), 1–60. | MR | Zbl

[28] S. D. Pauls, A notion of rectifiability modelled on Carnot groups, Indiana Univ. Math. J. 53 (2004), 49–81. | MR | Zbl

[29] D. Preiss, Geometry of Measures in ${ℝ}^n$: distribution, rectifiability and densities, Ann. of Math. 125 (1987), 537–643. | MR | Zbl

[30] S. Rigot, Counter example to the isodiametric inequality in H-type groups, preprint (2004).

[31] L. Simon, “Lectures on Geometric Measure Theory”, Proceedings of the Centre for Mathematical Analysis, Australian National University, Vol. 3, 1983. | MR | Zbl

[32] E. M. Stein, “Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals”, Princeton University Press, Princeton, 1993. | MR | Zbl

[33] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis and Geometry on Groups”, Cambridge University Press, Cambridge, 1992. | MR | Zbl

[34] S. Wenger and R. Young, Lipschitz extensions into jet space Carnot group, Math. Res. Lett., to appear. | MR | Zbl