# Blow-up of regular submanifolds in Heisenberg groups and applications

Open Mathematics (2006)

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

## Access Full Article

top## Abstract

top## How to cite

topValentino 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] 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] 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] A. Bellaïche and J.J. Risler (Eds.): Sub-Riemannian geometry, Progress in Mathematics, Vol. 144, Birkhäuser Verlag, Basel, 1996.
- [4] Y.D. Burago and V.A. Zalgaller: Geometric inequalities, Grundlehren Math. Springer, Berlin. Zbl0436.52009
- [5] H. Federer: Geometric Measure Theory, Springer, 1969.
- [6] G.B. Folland and E.M. Stein: Hardy Spaces on Homogeneous groups, Princeton University Press, 1982. Zbl0508.42025
- [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] B. Franchi, R. Serapioni and F. Serra Cassano: “Rectifiability and Perimeter in the Heisenberg group”, Math. Ann., Vol. 321(3), (2001). Zbl1057.49032
- [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] B. Franchi, R. Serapioni and F. Serra Cassano: Regular submanifolds, graphs and area formula in Heisenberg groups, preprint, (2004). Zbl1125.28002
- [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] 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] P. Hajlasz and P. Koskela: “Sobolev met Poincaré”, Mem. Amer. Math. Soc., Vol. 145, (2000). Zbl0954.46022
- [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] 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] I. Kupka: “Géométrie sous-riemannienne”, Astérisque, Vol. 241(817,5), (1997), pp. 351–380.
- [17] V. Magnani: “Differentiability and Area formula on stratified Lie groups”, Houston Jour. Math., Vol. 27(2), (2001), pp. 297–323. Zbl0983.22009
- [18] V. Magnani: “On a general coarea inequality and applications”, Ann. Acad. Sci. Fenn. Math., Vol. 27, (2002), pp. 121–140. Zbl1064.49034
- [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] 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] V. Magnani: “Note on coarea formulae in the Heisenberg group”, Publ. Mat., Vol. 48(2), (2004), pp. 409–422. Zbl1062.22020
- [22] V. Magnani: “Characteristic points, rectifiability and perimeter measure on stratified groups”, J. Eur. Math. Soc., to appear. Zbl1107.22004
- [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] P.Pansu, Geometrie du Group d'Heisenberg, Thesis (PhD), 3rd ed., Université Paris VII, 1982.
- [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] 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] E.M. Stein: Harmonic Analysis, Princeton University Press, 1993.
- [28] N.Th. Varopoulos, L. Saloff-Coste and T. Coulhon: Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992. Zbl0813.22003

## NotesEmbed ?

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