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 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.
@article{ASNSP_2010_5_9_4_687_0, author = {Mattila, Pertti and Serapioni, Raul and Serra Cassano, Francesco}, title = {Characterizations of intrinsic rectifiability in {Heisenberg} groups}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {687--723}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {4}, year = {2010}, mrnumber = {2789472}, zbl = {1229.28004}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_687_0/} }
TY - JOUR AU - Mattila, Pertti AU - Serapioni, Raul AU - Serra Cassano, Francesco TI - Characterizations of intrinsic rectifiability in Heisenberg groups JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 687 EP - 723 VL - 9 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_4_687_0/ LA - en ID - ASNSP_2010_5_9_4_687_0 ER -
%0 Journal Article %A Mattila, Pertti %A Serapioni, Raul %A Serra Cassano, Francesco %T Characterizations of intrinsic rectifiability in Heisenberg groups %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 687-723 %V 9 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_4_687_0/ %G en %F ASNSP_2010_5_9_4_687_0
Mattila, Pertti; Serapioni, Raul; Serra Cassano, Francesco. Characterizations of intrinsic rectifiability in Heisenberg groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 687-723. http://www.numdam.org/item/ASNSP_2010_5_9_4_687_0/
[1] “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Oxford University Press, 2000. | MR | Zbl
, and ,[2] Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527–555. | MR | Zbl
and ,[3] Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal. 19 (2009), 509–540. | MR | Zbl
, and ,[4] Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal. 16 (2006), 187–232. | MR | Zbl
, and ,[5] Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs, Calc. Var. Partial Differential Equations, 35 (2009), 17–49. | MR | Zbl
and ,[6] Rectifiability and Lipschitz extensions into the Heisenberg group, Math. Z. 263 (2009), 673–683. | MR | Zbl
and ,[7] Comparison of Hausdorff measures with respect to the Heisenberg metric, Publ. Mat. 47 (2003), 237–259. | EuDML | MR | Zbl
, and ,[8] “Stratified Lie Groups and Potential Theory for their Sub–Laplacians”, Springer Monographs in Mathematics, Springer-Verlag, 2007. | MR | Zbl
, and ,[9] “An Introduction to the Heisenberg Group and the sub-Riemannian Isoperimetric Problem”, Birkhäuser, 2007. | MR | Zbl
, , and ,[10] Differentiating maps into and the geometry of functions, Ann. of Math. 171 (2010), 1347–1385. | MR | Zbl
and ,[11] Implicit function theorem in Carnot Carathéodory spaces, Commun. Contemp. Math. 8, (2006), 253–293. | MR | Zbl
and ,[12] “Geometric Measure Theory”, Springer-Verlag, 1969. | MR | Zbl
,[13] “Hardy spaces in Homogeneous Groups”, Princeton University Press, 1982.
and ,[14] Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479–531. | MR | Zbl
, and ,[15] On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003), 421–466. | MR | Zbl
, and ,[16] Regular hypersurfaces, intrinsic perimeter and implicit function theorem, Comm. Anal. Geom. 11 (2003), 909–944. | MR | Zbl
, and ,[17] Regular submanifolds, graphs and area formula in Heisenberg Groups, Adv. Math. 211 (2007), 152–203. | MR | Zbl
, and ,[18] Differentiability of intrinsic Lipschitz functions within Heisenberg groups, J. Geom. Anal. DOI 10:1007/s12220-010-9178. | MR
, and ,[19] 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] Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 871–896. | EuDML | Numdam | MR | Zbl
and ,[21] “Elements of Geometric Measure Theory on Sub-Riemannion groups”, Edizioni della Normale, Pisa, 2002. | MR
,[22] Contact equations, Lipschitz extensions and isoperimetric inequalities, preprint http://cvgmt.sns.it/cgi/get.cgi/papers/maga/ (2009). | MR
,[23] “Geometry of Sets and Measures in Euclidean Spaces”, Cambridge University Press, 1995. | MR
,[24] Measures with unique tangent measures in metric groups, Math. Scand. 97 (2005), 298–308. | MR | Zbl
,[25] On Carnot-Carathèodory metrics, J. Differential Geom. 21 (1985), 35–45. | MR | Zbl
,[26] “A Tour of Sub-Riemannian Geometries, their Geodesics and Applications”, Mathematical Survey and Monographs, Vol. 91, American Mathematical Society, Providence RI, 2002. | MR
,[27] Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989), 1–60. | MR | Zbl
,[28] A notion of rectifiability modelled on Carnot groups, Indiana Univ. Math. J. 53 (2004), 49–81. | MR | Zbl
,[29] Geometry of Measures in : distribution, rectifiability and densities, Ann. of Math. 125 (1987), 537–643. | MR | Zbl
,[30] Counter example to the isodiametric inequality in H-type groups, preprint (2004).
,[31] “Lectures on Geometric Measure Theory”, Proceedings of the Centre for Mathematical Analysis, Australian National University, Vol. 3, 1983. | MR | Zbl
,[32] “Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals”, Princeton University Press, Princeton, 1993. | MR | Zbl
,[33] “Analysis and Geometry on Groups”, Cambridge University Press, Cambridge, 1992. | MR | Zbl
, and ,[34] Lipschitz extensions into jet space Carnot group, Math. Res. Lett., to appear. | MR | Zbl
and ,