VARIATION OF PERIMETER MEASURE IN SUB-RIEMANNIAN GEOMETRY

Year 2013, Volume: 6 Issue: 1, 8 - 40, 30.04.2013

Robert K. Hladky Scott D. Pauls

Abstract

Keywords

Carnot-Caratheodory, minimal surfaces, CMC surfaces, isoperimetric problem, characteristic set

References

• [1] V. Barone Adesi and F. Serra Cassano and D. Vittone, The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations, Calc. Var and PDE, 30(2007) no.1 , 17-49.
• [2] Z. Balogh, Size of Characteristic Sets and Functions with Prescribed Gradient., J. fr die Reine und Angewandte Mathematik, 564(2003), 63-83.
• [3] Bonk, M. and Capogna, L, Mean curvature flow in the Heisenberg group, 2005, preprint.
• [4] Bryant, R. and Griffiths, P. and Grossman, D., Exterior differential systems and Euler- Lagrange partial differential equations, Chicago Lectures in Mathematics, Chicago, IL, 2003.
• [5] Capogna, L. and Danielli, D. and Pauls, S. and Tyson, J., An introduction to the Heisen- berg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, 259, Birkhauser, Basel, 2007.
• [6] Cheng J.-H. and Hwang J.-F.and Malchiodi, A. and Yang, P., Minimal surfaces in pseudo- hermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(2005), no. 1, 129-177.
• [7] Cheng J.-H. and Hwang J.-F. and Yang, P., Existence and uniqueness of p-area minimizers in the Heisenberg group, Math. Ann. 337(2007), no.2, 253-293.
• [8] Citti, G. and Sarti, A., A cortical based model of perceptual completion in the Roto- Translation space, J. Math. Imaging Vis. 24(2006), 307326.
• [9] Cole, D., On minimal surfaces in Martinet-type spaces., Ph.D. Thesis, Dartmouth College, 2005.
• [10] Danielli, D. and Garofalo, N. and Nhieu, D.-M., Sub-Riemannian calculus on hypersurfaces in Carnot groups, Adv. Math 215(2007), no. 1, 292-378.
• [11] Danielli, D. and Garofalo, N. and Nhieu, D.-M., Minimal surfaces, surfaces of constant mean curvature and isoperimetry in Carnot groups, 2001, preprint.
• [12] , Danielli, D. and Garofalo, N. and Nhieu, D.-M., A notable family of entire intrinsic min- imal graphs in the Heisenberg group which are not perimeter minimizing, Amer. J. Math. 130(2008), no. 2, 317-339.
• [13] , Danielli, D. and Garofalo, N. and Nhieu, D.-M. and Pauls, S.,Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group , J. Diff. Geom. 81(2009), no. 2, 251-295.
• [14] Danielli, D. and Garofalo, N. and Nhieu, D.-M. and Pauls, S., Stable C2 complete embedded noncharacteristic H-minimal surfaces are vertical planes, preprint, arXiv:0903.4296 [math.DG].
• [15] Derridj, M., Sur un th´eor`eme de traces, Ann. de l’institut Fourier 22(1972), no. 2, 73-83.
• [16] do Carmo, M. and Peng, C. K.,Stable complete minimal surfaces in R3 are planes, Bull. Amer. Math. Soc. (N.S.) 1(1979), no. 6, 903-906.
• [17] Fischer-Colbrie, D. and Schoen, R., The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33(1980), no. 2, 199- 211.
• [18] Garofalo, N. and Pauls, S., The Bernstein problem in the Heisenberg group., preprint, arXiv:math/0209065 [math.DG].
• [19] Hirsch, M.W., Differential Topology, Springer-Verlag, 1997.
• [20] Hladky, R. and Pauls, S., Constant Mean Curvature Surfaces in sub-Riemannian Geometry, J. Diff. Geom. 79(2008), no. 1, 111-139.
• [21] Hladky, R., Connections and curvature in sub-Riemannian geometry, Houston J. Math. 38(2012), no. 4, 1107-1134.
• [22] Hoffman, W., The visual cortex is a contact bundle, Appl. Math. Comput. 32(1989), no. 2-3, 137-167.
• [23] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, John Wiley & Sons, Inc., 1963.
• [24] Leonardi, G. P. and Rigot, S., Isoperimetric sets on Carnot groups, Houston J. Math. 29(2003), no. 3, 609-637.
• [25] Magnani, V., Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc. 8(2006), no. 4, 585-609.
• [26] Magnani, V., Differentiability and Area Fomula on Stratified Lie groups, preprint.
• [27] Montefalcone,F., Hypersurfaces and variational formulas in sub-Riemannian Carnot groups, J. Math. Pures Appl. 87(2007), 453-494.
• [28] Monti, R. and Rickly, M., Convex isoperimetric sets in the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 391-415.
• [29] Pansu, P., M´etriques de Carnot-Carath´eodory et quasiisom´etries des espaces sym´etriques de rang un, Annals of Mathematics (2) 129(1989), no. 1, 1-60.
• [30] Pauls, S., Minimal surfaces in the Heisenberg group., Geom. Ded. 104(2004), 201-231.
• [31] Petitot, J., The neurogeometry of pinwheels as a sub-Riemannian contact structure, J. Phys- iology 97(2003), 265–309.
• [32] J. Petitot and Y. Tondut, Vers une neuro-geometrie. Fibrations corticales, structures de contact et contours subjectifs modaux., Math., Info. et Sc. Hum., EHESS, Paris 145(1998), 5–101.
• [33] Ritor´e, M. and Rosales, C., Area-stationary surfaces in the Heisenberg group H1, Mat. Con- temp. 35 (2008), 185-203.
• [34] Ritor´e, M. and Rosales, C., Rotationally invariant hypersurfaces with constant mean curva- ture in the Heisenberg group Hn, J. Geom. Anal. 16 (2006), no. 4, 703720.
• [35] Selby, C., Geometry of hypersurfaces in Carnot groups of step two., Ph.D. Thesis, Purdue University, 2006.
• [36] Shcherbakova, N., Minimal surfaces in contact sub-Riemannian manifolds, preprint, arXiv:math/0604494 [math.DG].
• [37] N. Tanaka, A differential geometric study on strongly pseudoconvex manifolds., Kinokuniya Book-Store Co., Ltd., 1975.
• [38] S.M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Diff. Geom., 13(1978), 25-41.