Convex isoperimetric sets in the Heisenberg group

Monti, Roberto; Rickly, Matthieu

Monti, Roberto; Rickly, Matthieu

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 2, p. 391-415

Abstract

We characterize convex isoperimetric sets in the Heisenberg group. We first prove Sobolev regularity for a certain class of $ℝ^{2}$ -valued vector fields of bounded variation in the plane related to the curvature equations. Then we show that the boundary of convex isoperimetric sets is foliated by geodesics of the Carnot-Carathéodory distance.

MR 2548252 | Zbl 1170.49037

Classification: 49Q20, 53C17

@article{ASNSP_2009_5_8_2_391_0,
     author = {Monti, Roberto and Rickly, Matthieu},
     title = {Convex isoperimetric sets in the Heisenberg group},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {2},
     year = {2009},
     pages = {391-415},
     zbl = {1170.49037},
     mrnumber = {2548252},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_2_391_0}
}

Monti, Roberto; Rickly, Matthieu. Convex isoperimetric sets in the Heisenberg group. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 2, pp. 391-415. http://www.numdam.org/item/ASNSP_2009_5_8_2_391_0/

References

[1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), 227–260. | MR 2096794 | Zbl 1075.35087

[2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Oxford Science Publications, Clarendon Press, Oxford, 2000. | MR 1857292 | Zbl 0957.49001

[3] L. Capogna, D. Danielli, D. P. Scott and J. T. Tyson, “An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem”, Birkhäuser, 2007. | MR 2312336 | Zbl 1138.53003

[4] J.-H. Cheng, J.-F. Hwang and P. Yang, Existence and uniqueness for $p$ -area minimizers in the Heisenberg group, Math. Ann. 337 (2007), 253–293. | MR 2262784 | Zbl 1109.35009

[5] J.-H. Cheng, J.-F. Hwang and P. Yang, Regularity of $C^{1}$ smooth surfaces with prescribed $p$ -mean curvature in the Heisenberg group, preprint, 2007. | MR 2481053

[6] J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 129–177. | Numdam | MR 2165405 | Zbl 1158.53306

[7] D. Danielli, N. Garofalo and D.-M. Nhieu, A partial solution of the isoperimetric problem for the Heisenberg group, Forum Math. 20 (2008), 99–143. | MR 2386783 | Zbl 1142.53021

[8] B. Franchi, S. Gallot and R. L. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557–571. | MR 1314734 | Zbl 0830.46027

[9] B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479–531. | MR 1871966 | Zbl 1057.49032

[10] N. Garofalo and 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. | MR 1404326 | Zbl 0880.35032

[11] G. P. Leonardi and S. Masnou, On the isoperimetric problem in the Heisenberg group $ℍ^{n}$ , Ann. Mat. Pura Appl. (4) 184 (2005), 533–553. | MR 2177813 | Zbl 1223.49051

[12] G. P. Leonardi and S. Rigot, Isoperimetric sets on Carnot groups, Houston J. Math. 29 (2003), 609–637. | MR 2000099 | Zbl 1039.49037

[13] R. Monti, Brunn-Minkowski and isoperimetric inequality in the Heisenberg group, Ann. Acad. Sci. Fenn. Math. 28 (2003), 99–109. | MR 1976833 | Zbl 1075.49017

[14] R. Monti, Heisenberg isoperimetric problem. The axial case, Adv. Calc. Var. 1 (2008), 93–121. | MR 2402213 | Zbl 1149.49036

[15] R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane, J. Geom. Anal. 14 (2004), 355–368. | MR 2051692 | Zbl 1076.53035

[16] R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), 339–376. | MR 1865002 | Zbl 1032.49045

[17] P. Pansu, Une inégalité isopérimétrique sur le groupe de Heisenberg, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 127–130. | MR 676380 | Zbl 0502.53039

[18] P. Pansu, An isoperimetric inequality on the Heisenberg group, Conference on differential geometry on homogeneous spaces (Turin, 1983). Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1983), 159–174. | MR 829003 | Zbl 0634.53034

[19] S. D. Pauls, Minimal surfaces in the Heisenberg group, Geom. Dedicata 104 (2004), 201–231. | MR 2043961 | Zbl 1054.49029

[20] S. D. Pauls, H-minimal graphs of low regularity in $ℍ^{1}$ , Comment. Math. Helv. 81 (2006), 337–381. | MR 2225631 | Zbl 1153.53305

[21] M. Ritoré, A proof by calibration of an isoperimetric inequality in the Heisenberg group, 2008, preprint. | MR 2898770

[22] M. Ritoré and C. Rosales, Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group $ℍ^{n}$ . J. Geom. Anal. 16 (2006), 703–720. | MR 2271950 | Zbl 1129.53041

[23] M. Ritoré and C. Rosales, Area-stationary surfaces in the Heisenberg group $ℍ^{1}$ , Adv. Math. 219 (2008), 633–671. | MR 2435652 | Zbl 1158.53022