Abstract
We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot–Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot–Carathéodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.
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Balogh, Z.: Size of characteristic sets and functions with prescribed gradient, To appear in J. Reine Angew. Math.
Bernstein, S.: Sur un théorè me de géométrie et ses applications aux équations aux déricées partielles du type elliptique, Comm. de la Soc. Math. de Kharkov (2-éme sér) 15 (1915), 38-45.
Capogna, L., Danielli, D. and Garfola, N.: An isoperimetric inequality and the geometric Sobolev embedding for vector fields, Math. Res. Lett. 1(2) (1994), 263-268.
Capogna, L., Danielli, D. and Garfola, N.: The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2(2) (1994), 203-215.
DeGiorgi, E.: Su una teoria generale della misura (r-1) dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. 4(36) (1954), 191-213.
[DGN01] Danielli, D., Garofalo, N. and Nhieu, D. M.: Minimal surfaces, surfaces of constant mean curvature and isoperimetry in Carnot groups, Preprint 2001.
Figueroa, C. B., Mercuri, F. and Pedrosa, R. H. L.: Invariant surfaces of the Heisenberg groups, Ann. Mat. Pura Appl. (4) 177 (1999), 173-194.
Franchi, B., Serapioni, R. and Serra Cassano, F.: Champs de vecteurs, théoréme d'approximation de Meyers-Serrin et phénomè ne de Lavrentev pour des fonctionnelles dégénérées, C.R. Acad. Sci. Paris Sér. I Math 320(6) (1995), 695-698.
Franchi, B., Serapioni, R. and Serra Cassano, F.: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math 22(4) (1996), 859-890.
Franchi, B., Serapioni, R. and Serra Cassano, F.: Rectifiability and Perimeter in the Heisenberg group, Math. Ann. 321(3) (2001), 479-531.
Garofalo, N. and Nhieu, Duy-Minh: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49(10) (1996), 1081-1144.
Garofalo, N. and Pauls, S. D.: The Bernstein problem in the Heisenberg group, preprint 2002.
Gromov, M.: Structures métriques pour les variétés riemanniennes, In: J. Lafontaine and P. Pansu (eds), Textes mathématiques 1 CEDIC, Paris, 1981.
Gromov, M.: Groups of polynomial growth and expanding maps, I.H.E.S. Publ. Math. (53) (1981), 53-73.
Gromov, M.: Carnot-Carathéodory spaces seen from within, In: Sub-Riemannian Geometry, Progr. Math. 144, Birkhaüser, Basel, 1996, pp. 79-323.
Gilbarg, D. and Trudinger, N. S.: Elliptic Partial Differential Equations of Second order, Springer-Verlag, Berlin, 2001.
Heinonen, J.: Calculus on Carnot groups, In: Fall School in Analysis (Jyväskylä, 1994), Univ. Jyväskylä, Jyväskylä, 1995, pp. 1-31.
Hsiang, Wu-teh and Hsiang, Wu-yi.: On the existence of codimension-one minimal spheres in compact symmetric spaces of rank 2. II, J. Differential Geom. 17(4) (1983, 1982), 583-594.
Hsiang, Wu-yi and Lawson, B. H. Jr.: Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1-38.
Monti, R. and Cassano, F. S.: Surface measures in Carnot-Carathé odory spaces, Preprint 1999.
Pansu, P.: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129(1) (1989), 1-60.
per Tomter: Constant mean curvature surfaces in the Heisenberg group, In: Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Amer. Math. Soc., Providence, RI, 1993, pp. 485-495.
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Pauls, S.D. Minimal Surfaces in the Heisenberg Group. Geometriae Dedicata 104, 201–231 (2004). https://doi.org/10.1023/B:GEOM.0000022861.52942.98
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DOI: https://doi.org/10.1023/B:GEOM.0000022861.52942.98