Blow-up of regular submanifolds in Heisenberg groups and applications

Valentino Magnani

Open Mathematics (2006)

  • Volume: 4, Issue: 1, page 82-109
  • ISSN: 2391-5455

Abstract

top
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.

How to cite

top

Valentino Magnani. "Blow-up of regular submanifolds in Heisenberg groups and applications." Open Mathematics 4.1 (2006): 82-109. <http://eudml.org/doc/268872>.

@article{ValentinoMagnani2006,
abstract = {We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.},
author = {Valentino Magnani},
journal = {Open Mathematics},
keywords = {28A75; 22E25},
language = {eng},
number = {1},
pages = {82-109},
title = {Blow-up of regular submanifolds in Heisenberg groups and applications},
url = {http://eudml.org/doc/268872},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Valentino Magnani
TI - Blow-up of regular submanifolds in Heisenberg groups and applications
JO - Open Mathematics
PY - 2006
VL - 4
IS - 1
SP - 82
EP - 109
AB - We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.
LA - eng
KW - 28A75; 22E25
UR - http://eudml.org/doc/268872
ER -

References

top
  1. [1] L. Ambrosio: “Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces”, Adv. Math., Vol. 159, (2001), pp. 51–67. http://dx.doi.org/10.1006/aima.2000.1963 
  2. [2] Z.M. Balogh: “Size of characteristic sets and functions with prescribed gradients”, J. Reine Angew. Math., Vol. 564, (2003), pp. 63–83. Zbl1051.53024
  3. [3] A. Bellaïche and J.J. Risler (Eds.): Sub-Riemannian geometry, Progress in Mathematics, Vol. 144, Birkhäuser Verlag, Basel, 1996. 
  4. [4] Y.D. Burago and V.A. Zalgaller: Geometric inequalities, Grundlehren Math. Springer, Berlin. Zbl0436.52009
  5. [5] H. Federer: Geometric Measure Theory, Springer, 1969. 
  6. [6] G.B. Folland and E.M. Stein: Hardy Spaces on Homogeneous groups, Princeton University Press, 1982. Zbl0508.42025
  7. [7] B. Franchi, R. Serapioni and F. Serra Cassano: “Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields”, Houston Jour. Math., Vol. 22, (1996), pp. 859–889. Zbl0876.49014
  8. [8] B. Franchi, R. Serapioni and F. Serra Cassano: “Rectifiability and Perimeter in the Heisenberg group”, Math. Ann., Vol. 321(3), (2001). Zbl1057.49032
  9. [9] B. Franchi, R. Serapioni and F. Serra Cassano: “Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups”, Comm. Anal. Geom., Vol. 11(5), (2003), pp. 909–944. Zbl1077.22008
  10. [10] B. Franchi, R. Serapioni and F. Serra Cassano: Regular submanifolds, graphs and area formula in Heisenberg groups, preprint, (2004). Zbl1125.28002
  11. [11] M. Gromov: “Carnot-Carathéodory spaces seen from within”, In: A. Bellaiche and J. Risler (Eds.): Subriemannian Geometry, Progress in Mathematics, Vol. 144, Birkhauser Verlag, Basel, 1996. Zbl0864.53025
  12. [12] N. Garofalo and D.M. Nhieu: “Isoperimetric and Sobolev Inequalities for Carnot-Carathéodory Spaces and the Existence of Minimal Surfaces”, Comm. Pure Appl. Math., Vol. 49, (1996), pp. 1081–1144. http://dx.doi.org/10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A 
  13. [13] P. Hajlasz and P. Koskela: “Sobolev met Poincaré”, Mem. Amer. Math. Soc., Vol. 145, (2000). Zbl0954.46022
  14. [14] B. Kirchheim and F. Serra Cassano: “Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. 3(4), (2004), pp. 871–896. Zbl1170.28300
  15. [15] A. Korányi: “Geometric properties of Heisenberg-type groups”, Adv. Math., Vol. 56(1), (1985), pp. 28–38. http://dx.doi.org/10.1016/0001-8708(85)90083-0 
  16. [16] I. Kupka: “Géométrie sous-riemannienne”, Astérisque, Vol. 241(817,5), (1997), pp. 351–380. 
  17. [17] V. Magnani: “Differentiability and Area formula on stratified Lie groups”, Houston Jour. Math., Vol. 27(2), (2001), pp. 297–323. Zbl0983.22009
  18. [18] V. Magnani: “On a general coarea inequality and applications”, Ann. Acad. Sci. Fenn. Math., Vol. 27, (2002), pp. 121–140. Zbl1064.49034
  19. [19] V. Magnani: “A Blow-up Theorem for regular hypersurfaces on nilpotent groups”, Manuscripta Math., Vol. 110(1), (2003), pp. 55–76. http://dx.doi.org/10.1007/s00229-002-0303-y Zbl1010.22010
  20. [20] V. Magnani: “The coarea formula for real-valued Lipschitz maps on stratified groups”, Math. Nachr., Vol. 278(14), (2005), pp. 1–17. http://dx.doi.org/10.1002/mana.200310334 
  21. [21] V. Magnani: “Note on coarea formulae in the Heisenberg group”, Publ. Mat., Vol. 48(2), (2004), pp. 409–422. Zbl1062.22020
  22. [22] V. Magnani: “Characteristic points, rectifiability and perimeter measure on stratified groups”, J. Eur. Math. Soc., to appear. Zbl1107.22004
  23. [23] R. Montgomery: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, 2002. Zbl1044.53022
  24. [24] P.Pansu, Geometrie du Group d'Heisenberg, Thesis (PhD), 3rd ed., Université Paris VII, 1982. 
  25. [25] P. Pansu, “Une inégalité isoperimetrique sur le groupe de Heisenberg”, C.R. Acad. Sc. Paris, Série I, Vol. 295, (1982), pp. 127–130. Zbl0502.53039
  26. [26] P. Pansu: “Métriques de Carnot-Carathéodory quasiisométries des espaces symétriques de rang un”, Ann. Math., Vol. 129, (1989), pp. 1–60. http://dx.doi.org/10.2307/1971484 Zbl0678.53042
  27. [27] E.M. Stein: Harmonic Analysis, Princeton University Press, 1993. 
  28. [28] N.Th. Varopoulos, L. Saloff-Coste and T. Coulhon: Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992. Zbl0813.22003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.