Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group

 Access to full text

 Full (PDF)

1. [1] L. Ambrosio – B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527-555. Zbl0966.28002MR1800768
2. [2] L. Ambrosio – B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80. Zbl0984.49025MR1794185
3. [3] L. Ambrosio – V. Magnani, Weak differentiability of $BV$ function on stratified groups, Math. Z. 245 (2003), 123-153. Zbl1048.49030MR2023957
4. [4] Z. Balogh, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math. 564 (2003), 63-84. Zbl1051.53024MR2021034
5. [5] Z. Balogh – M. Rickly – F. Serra Cassano, Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric, Publ. Mat. 47 (2003), 237-259. Zbl1060.28002MR1970902
6. [6] Z. Balogh – H. Hofer-Isenegger – J. T. Tyson, Lifts of Lipischitz maps and horizontal fractals in the Heisenberg group, Preprint (2003). 
7. [7] A. Bellaïche, “The tangent space in subriemannian geometry”, In: “Subriemannian Geometry", Progress in Mathematics 144, A. Bellaiche – J. Risler (eds.), Birkhauser Verlag, Basel, 1996. Zbl0862.53031MR1421822
8. [8] G. Bellettini – M. Paolini – S. Venturini, Some results in surface measure in Calculus of Variations, Ann. Mat. Pura Appl. (4) 170 (1996), 329-357. Zbl0890.49020MR1441625
9. [9] M. Biroli – U. Mosco, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1995), 37-44. Zbl0837.31006MR1340280
10. [10] L. Capogna – D. Danielli – N. Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 12 (1994), 203-215. Zbl0864.46018MR1312686
11. [11] T. Coulhon – L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), 293-314. Zbl0782.53066MR1232845
12. [12] M. Chlebík, Hausdorff lower $s$-densities and rectifiability of sets in $n$-space, Preprint. 
13. [13] D. Danielli – N. Garofalo – D. M. Nhieu, Traces inequalities for Carnot–Carathèodory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 195-252. Zbl0938.46036MR1664688
14. [14] G. David – S. Semmes, “Fractured Fractals and Broken Dreams. Self-Similar Geometry through Metric and Measure”, Oxford University Press, 1997. Zbl0887.54001MR1616732
15. [15] G. David – T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann. 315 (1999), 641-710. Zbl0944.53004MR1731465
16. [16] E. De Giorgi, Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni, Ann.Mat.Pura Appl. (4) 36 (1954), 191-213. Zbl0055.28504MR62214
17. [17] E. De Giorgi, Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni, Ricerche Mat. 4 (1955), 95-113. Zbl0066.29903MR74499
18. [18] E. De Giorgi – F. Colombini – L. C. Piccinini, “Frontiere orientate di misura minima e questioni collegate”, Scuola Normale Superiore, Pisa, 1972. Zbl0296.49031MR493669
19. [19] E. De Giorgi, Problema di Plateau generale e funzionali geodetici, Atti Sem. Mat. Fis. Univ. Modena 43 (1995), 285-292. Zbl0862.49028MR1366062
20. [20] E. De Giorgi, Un progetto di teoria unitaria delle correnti, forme differenziali, varietà ambientate in spazi metrici, funzioni a variazione limitata, Manuscript, (1995). 
21. [21] E. De Giorgi, Un progetto di teoria delle correnti, forme differenziali e varietà non orientate in spazi metrici, In: “Variational Methods, Non Linear Analysys and Differential Equations in Honour of J. P. Cecconi”, (Genova 1993), M. Chicco et al. (eds.), ECIG, Genova, 67-71. 
22. [22] H. Federer, “Geometric Measure Theory”, Springer, 1969. Zbl0176.00801MR257325
23. [23] B. Franchi – S. Gallot – R. L. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557-571. Zbl0830.46027MR1314734
24. [24] B. Franchi – R. Serapioni – F. Serra Cassano, Meyers-Serrin Type Theorems and Relaxation of Variational Integrals Depending Vector Fields, Houston J. Math. 22 (1996), 859-889. Zbl0876.49014MR1437714
25. [25] B. Franchi – R. Serapioni – F. Serra Cassano, Sur les ensembles des périmètre fini dans le groupe de Heisenberg, C.R. Acad. Sci. Paris Ser. I Math. 329 (1999), 183-188. Zbl1033.49045MR1711057
26. [26] B. Franchi – R. Serapioni – F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479-531. Zbl1057.49032MR1871966
27. [27] B. Franchi – R. Serapioni – F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom. 11 (2003) 909-944. Zbl1077.22008MR2032504
28. [28] B. Franchi – R. Serapioni – F. Serra Cassano, Rectifiability and perimeter in step 2 groups, Proceedings of Equadiff10, 2001, Math. Bohem. 127 (2002), 219-228. Zbl1018.49029MR1981527
29. [29] B. Franchi – R. Serapioni – F. Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003), 421-466. Zbl1064.49033MR1984849
30. [30] N. Garofalo – D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144. Zbl0880.35032MR1404326
31. [31] M. Gromov, “Carnot-Carathéodory spaces seen from within”, In: “Subriemannian Geometry", Progress in Mathematics 144, A. Bellaiche and J. Risler (eds.), Birkhauser Verlag, Basel, 1996. Zbl0864.53025MR1421823
32. [32] P. Hajłasz – P. Koskela, “Sobolev met Poincare”, Mem. AMS 145, 2000. Zbl0954.46022MR1683160
33. [33] J. Heinonen – P. Koskela – N. Shanmuganlingam – J. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87-139. Zbl1013.46023MR1869604
34. [34] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113-123. Zbl0806.28004MR1189747
35. [35] B. Kirchheim – V. Magnani, A counterexample to the metric differentiability, Proc. Edinburgh Math. Soc. 46 (2003), 221-227. Zbl1059.26007MR1961822
36. [36] A. Korányi – H. M. Reimann, Foundation for the Theory of Quasiconformal Mappings on the Heisenberg Group, Adv. Math. 111 (1995), 1-87. Zbl0876.30019
37. [37] A. Lorent, Rectifiability of measures with locally uniform cube density, Proc. London Math. Soc. (3) 86 (2003), 153-249. Zbl1042.28002MR1971467
38. [38] V. Magnani, Differentiability and Area formula on stratified Lie groups, Houston J. Math. 27 (2001), 297-323. Zbl0983.22009MR1874099
39. [39] V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, Preprint (2003). Zbl1107.22004MR2262196
40. [40] M. A. Martin – P. Mattila, Hausorff measures, Hölder continuous maps and self-similar fractals, Math. Proc. Cambridge Philos. Soc. 114 (1993), 37-42. Zbl0783.28005MR1219912
41. [41] M. A. Martin – P. Mattila, On The Parametrization of Self-Similar And Other Fractal Sets, Trans. Amer. Math. Soc. 128 (2000), 2641-2648. Zbl0951.28005MR1664402
42. [42] P. Mattila, “Geometry of sets and measures in Euclidean spaces”, Cambridge U.P., 1995. Zbl0819.28004MR1333890
43. [43] J. Mitchell, On Carnot-Carathèodory metrics, J. Differential Geom. 21 (1985), 35-45. Zbl0554.53023MR806700
44. [44] R. Monti – F. Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), 339-376. Zbl1032.49045MR1865002
45. [45] A. Nagel – E. M. Stein – S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 103-147. Zbl0578.32044MR793239
46. [46] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989), 1-60. Zbl0678.53042MR979599
47. [47] P. Pansu, Une inégalité isopérimétrique sur le groupe de Heisenberg, C.R. Acad. Sci. Paris 295 I (1982), 127-130. Zbl0502.53039MR676380
48. [48] P. Pansu, Geometrie du Group d’Heisenberg, These pour le titre de Docteur 3ème cycle, Universite Paris VII, (1982). 
49. [49] S. D. Pauls, A notion of rectifiability modelled on Carnot groups, Indiana Univ. Math. J. 53 (2004), 49-81. Zbl1076.49025MR2048183
50. [50] D. Preiss – J. Tišer, On Besicovitch $1/2$-problem, J. London Math. Soc. 45 (1992), 279-287. Zbl0762.28003
51. [51] E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 198-223. Zbl0099.08503MR114145
52. [52] S. Semmes, Chord- arc surfaces with small constant I, Adv. Math. 85 (1991), 198-223. Zbl0733.42015MR1093006
53. [53] S. Semmes, Chord- arc surfaces with small constant II, Adv. Math. 88 (1991), 170-189. Zbl0733.42016MR1120612
54. [54] S. Semmes, On the non existence of bilipschitz parameterization and geometric problems about $A_{\infty }$ weights, Rev. Mat. Iberoamericana 12 (1996), 337-410. Zbl0858.46017
55. [55] S. Semmes, Good metric spaces without good parameterization, Rev. Mat. Iberoamericana 12 (1996), 187-275. Zbl0854.57018MR1387590
56. [56] L. Simon, “Lectures on Geometric Measure Theory”, Proc. Centre for Math. Anal., Australian Nat. Univ. 3, 1983. Zbl0546.49019MR756417
57. [57] E. M. Stein, “Harmonic Analysis”, Princeton University Press, 1993. Zbl0821.42001MR1232192
58. [58] R. Strichartz, Self-similarity on nilpotent Lie groups, Contemp. Math. 140 (1992), 123-157. Zbl0797.43004MR1197594
59. [59] T. Toro, Geometric conditions and existence of bi-lipschitz parameterizations, Duke Math. J. 77 (1995), 193-227. Zbl0847.42011MR1317632
60. [60] N. Th. Varopoulos, Analysis on Lie Groups, J. Funct. Anal. 76 (1988), 346-410. Zbl0634.22008MR924464
61. [61] N. Th. Varopoulos – L. Saloff-Coste – T. Coulhon, “Analysis and Geometry on Groups”, Cambridge University Press, Cambridge, 1992. Zbl1179.22009MR1218884
62. [62] S. K. Vodop’yanov, $𝒫$-differentiability on Carnot groups in different topologies and related topics, Proc. on Analysis and Geometry, 603-670, Sobolev Institute Press, Novosibirsk, 2000. Zbl0992.58005MR1847541

1. Bruno Franchi, Marco Marchi, Raul Paolo Serapioni, Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem
2. Valentino Magnani, Blow-up of regular submanifolds in Heisenberg groups and applications