Some properties of Carnot-Carathéodory balls in the Heisenberg group

Roberto Monti

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: 3, page 155-167
  • ISSN: 1120-6330

Abstract

top
Using the exact representation of Carnot-Carathéodory balls in the Heisenberg group, we prove that: 1. ∇ H n ⁡ d ⁢ z , t = 1 in the classical sense for all z , t ∈ H n with z ≠ 0 , where d is the distance from the origin; 2. Metric balls are not optimal isoperimetric sets in the Heisenberg group.

How to cite

top

Monti, Roberto. "Some properties of Carnot-Carathéodory balls in the Heisenberg group." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.3 (2000): 155-167. <http://eudml.org/doc/252408>.

@article{Monti2000,
abstract = {Using the exact representation of Carnot-Carathéodory balls in the Heisenberg group, we prove that: 1. $|\nabla_\{\mathbb\{H\}^\{n\}\} d(z,t)| = 1$ in the classical sense for all $(z,t) \in \mathbb\{H\}^\{n\}$ with $z \neq 0$, where $d$ is the distance from the origin; 2. Metric balls are not optimal isoperimetric sets in the Heisenberg group.},
author = {Monti, Roberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Isoperimetric inequality; Heisenberg group; Eikonal equation},
language = {eng},
month = {9},
number = {3},
pages = {155-167},
publisher = {Accademia Nazionale dei Lincei},
title = {Some properties of Carnot-Carathéodory balls in the Heisenberg group},
url = {http://eudml.org/doc/252408},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Monti, Roberto
TI - Some properties of Carnot-Carathéodory balls in the Heisenberg group
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/9//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 3
SP - 155
EP - 167
AB - Using the exact representation of Carnot-Carathéodory balls in the Heisenberg group, we prove that: 1. $|\nabla_{\mathbb{H}^{n}} d(z,t)| = 1$ in the classical sense for all $(z,t) \in \mathbb{H}^{n}$ with $z \neq 0$, where $d$ is the distance from the origin; 2. Metric balls are not optimal isoperimetric sets in the Heisenberg group.
LA - eng
KW - Isoperimetric inequality; Heisenberg group; Eikonal equation
UR - http://eudml.org/doc/252408
ER -

References

top
  1. Barbu, V., Mathematical methods in optimization of differential systems. Kluwer, Dordrecht1994. Zbl0819.49002MR1325922DOI10.1007/978-94-011-0760-0
  2. Bellaïche, A., The tangent space in subriemannian geometry. In: A. Bellaïche - J. Risler (eds), Sub-Riemannian Geometry. Progress in Mathematics, 144, Birkhäuser, Basel1996. Zbl0862.53031
  3. Bellettini, G. - Paolini, M. - Venturini, S., Some results in surface measure in Calculus of Variations. Ann. Mat. Pura Appl., (4), 170, 1996, 329-357. Zbl0890.49020MR1441625DOI10.1007/BF01758994
  4. Brockett, R.W., Control theory and singular Riemannian geometry. In: P.J. Hilton - G.S. Young, New directions in applied mathematics. Springer-Verlag, New York1981. Zbl0483.49035MR661282
  5. Yu, D. - Zalgaller, V.A., Geometric inequalities. Springer-Verlag, Berlin1988. Zbl0633.53002MR936419
  6. De Giorgi, E., Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Acc. Lincei Mem. fis., s. 8, v. 5, 1958, 33-44. Zbl0116.07901MR98331
  7. Folland, G.B. - Stein, E.M., Hardy spaces on homogeneous groups. Princeton University Press, 1982. Zbl0508.42025MR657581
  8. Franchi, B., Stime subellittiche e metriche riemanniane singolari II. Seminario di Analisi Matematica, Università di Bologna, 1983, VIII-1VIII-17. 
  9. Franchi, B. - Gallot, S. - Wheeden, R.L., Sobolev and isoperimetric inequalities for degenerate metrics. Math. Ann., 300, 1994, 557-571. Zbl0830.46027MR1314734DOI10.1007/BF01450501
  10. Franchi, B. - Serapioni, R. - Serra Cassano, F., Meyers-Serrin Type Theorems and Relaxation of Variational Integrals Depending Vector Fields. Houston Journal of Mathematics, 22, 4, 1996, 859-889. Zbl0876.49014MR1437714
  11. Franchi, B. - Serapioni, R. - Serra Cassano, F., Approximation and Imbedding Theorems for Weighted Sobolev Spaces Associated with Lipschitz Continuous Vector Fields. Bollettino U.M.I., 7, 11-B, 1997, 83-117. Zbl0952.49010MR1448000
  12. Franchi, B. - Serapioni, R. - Serra Cassano, F., Rectifiability and perimeter in the Heisenberg group. Preprint 2000. Zbl1057.49032MR1871966DOI10.1007/s002080100228
  13. Garofalo, N. - Nhieu, D.M., Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math., 49, 1996, 1081-1144. Zbl0880.35032MR1404326DOI10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
  14. Garofalo, N. - Nhieu, D.M., Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot Carathéodory spaces. Jour. Analyse Mathematique, 74, 1998, 67-97. Zbl0906.46026MR1631642DOI10.1007/BF02819446
  15. Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math., 139, 1977, 1-2, 95-153. Zbl0366.22010MR461589
  16. Gromov, M., Carnot-Carathéodory spaces seen from within. In: A. Bellaïche - J. Risler (eds.), Sub-Riemannian Geometry. Progress in Mathematics, 144, Birkhäuser, Basel1996. Zbl0864.53025MR1421823
  17. Gromov, M., Structures métriques pour les variétés riemanniennes. CEDIC, Paris1981. Zbl0509.53034MR682063
  18. Jerison, D. - Sanchez Calle, A., Subelliptic, second order differential operators. In: Complex analysis III. Springer, 1987, 46-77. Zbl0634.35017MR922334DOI10.1007/BFb0078245
  19. Liu, W. - Sussmann, H.J., Shortest paths for sub-Riemannian metrics on rank-two distributions. Mem. Amer. Math. Soc., 118, 1995. Zbl0843.53038MR1303093
  20. Montgomery, R., Abnormal minimizers. SIAM J. Control Optim., 32, 1994, 1605-1620. Zbl0816.49019MR1297101DOI10.1137/S0363012993244945
  21. Monti, R. - Serra Cassano, F., Surface measures in Carnot-Carathéodory spaces. Forthcoming. Zbl1032.49045
  22. Pansu, P., Une inégalité isopérimétrique sur le groupe de Heisenberg. C.R. Acad. Sci. Paris, 295, I, 1982, 127-130. Zbl0502.53039MR676380
  23. Stein, E.M., Harmonic Analysis. Princeton University Press, 1993. Zbl0821.42001MR1232192
  24. Strichartz, R.S., Sub-Riemannian geometry. J. Differential Geom., 24, 1986, 221-263.[Corrections to «Sub-Riemannian geometry». J. Differential Geom., 30, 1989, 595-596]. Zbl0609.53021MR862049

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.