Minimal surfaces in pseudohermitian geometry
Jih-Hsin Cheng[1]; Jenn-Fang Hwang[1]; Andrea Malchiodi[2]; Paul Yang[3]
- [1] Institute of Mathematics Academia Sinica Nankang, Taipei, Taiwan, 11529, R.O.C.
- [2] SISSA Via Beirut 2-4 34014 Trieste, Italy
- [3] Department of Mathematics Princeton University Princeton, NJ 08544, U.S.A.
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)
- Volume: 4, Issue: 1, page 129-177
- ISSN: 0391-173X
Access Full Article
topAbstract
topHow to cite
topReferences
top- [B] Z. Balogh, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math. 564 (2003), 63–83. Zbl1051.53024MR2021034
- [CDG] L. Capogna, D. Danielli and N. Garofalo, The geometric Sobolev embedding fir vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), 203–215. Zbl0864.46018MR1312686
- [CF] P. Concus and R. Finn, On capillary free surfaces in the absence of gravity, Acta Math. 132 (1974), 177–198. Zbl0382.76003MR670441
- [CH] J.-H. Cheng and J.-F. Hwang, Properly embedded and immersed minimal surfaces in the Heisenberg group, Bull. Austral. Math. Soc. 70 (2004), 507–520. Zbl1062.35046MR2103983
- [CK] P. Collin and R. Krust, Le problème de Dirichlet pour l’équation des surfaces minimales sur des domaines non bornès, Bull. Soc. Math. France 119 (1991), 443–462. Zbl0754.53013MR1136846
- [DGN] D. Danielli, N. Garofalo and D-M. Nhieu, Minimal surfaces, surfaces of constant mean curvature and isoperimetry in Carnot groups, Preprint, 2001.
- [FS] G. B. Folland and E. M. Stein, Estimates for the -complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. Zbl0293.35012MR367477
- [FSS] B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479–531. Zbl1057.49032MR1871966
- [GN] N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Caratheodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081–1144. Zbl0880.35032MR1404326
- [GP1] N. Garofalo and S. Pauls, The Bernstein problem in the Heisenberg group, arXiv: math. DG/0209065.
- [GP2] N. Garofalo and S. Pauls, The Bernstein problem in the Heisenberg group, preprint, 2004.
- [HL] R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47-157. Zbl0584.53021MR666108
- [Hw1] J. F. Hwang, Comparison principles and Liouville theorems for prescribed mean curvature equation in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1988), 341–355. Zbl0705.49022MR1015799
- [Hw2] J. F. Hwang, Structural inequalities method for uniqueness theorems for the minimal surface equation, Proc. of CMA (Joint Australia-Taiwan Workshop on Analysis and Application, Brisbane, March 1997), Australian National University, Vol. 37, 1999, 47–52. Zbl1193.35046
- [JL1] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167–197. Zbl0661.32026MR880182
- [JL2] D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), 303–343. Zbl0671.32016MR982177
- [Jo] F. John, “Partial Differential Equations”, Springer-Verlag, 4th ed., 1982. Zbl0472.35001MR831655
- [La] H. B. Lawson Jr., Complete minimal surfaces in , Ann. of Math. 92 (1970), 335-374. Zbl0205.52001MR270280
- [Lee] J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc. 296 (1986), 411–429. Zbl0595.32026MR837820
- [LM] G. Leonardi and S. Masnou, On the isoperimetric problem in the Heisenberg group , to appear on Annali Mat. Pura e Appl., 2002. Zbl1223.49051MR2177813
- [LR] G. Leonardi and S. Rigot, Isoperimetric sets on Carnot groups, Houston J. Math. (2003). Zbl1039.49037MR2000099
- [Mik] V. M. Miklyukov, On a new approach to Bernstein’s theorem and related questions for equations of minimal surface type, Mat. Sb. 108 (150) (1979), 268–289; English transl. in Math. USSR Sb. 36 (1980), 251–271. Zbl0488.49029MR525842
- [Mil] J. Milnor, “Topology from the Differentiable Viewpoint”, University of Virginia Press, 1965. Zbl0136.20402MR226651
- [Mo] G. Monge, “Application de l’Analyse à la Géométrie”, Paris, Bachelier, 1850.
- [Os] R. Osserman, “A Survey of Minimal Surfaces”, Dover Publications, Inc., New York, 1986. MR852409
- [Pan] P. Pansu, Une inegalite isoperimetrique sur le groupe de Heisenberg, C.R. Acad. Sci. Paris I 295 (1982), 127–130. Zbl0502.53039MR676380
- [Pau] S. D. Pauls, Minimal surfaces in the Heisenberg group, Geom. Dedicata 104 (2004), 201–231. Zbl1054.49029MR2043961
- [Sp] M. Spivak, “A Comprehensive Introduction to Differential Geometry”, Vol. 3, Publish or Perish Inc., Boston, 1975. Zbl0306.53001
- [St] S. Sternberg, “Lectures on Differential Geometry”, 2nd ed., Chelsea Publishing Company, New York, 1983. Zbl0518.53001MR891190
- [SY] R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45–76. Zbl0405.53045MR526976
- [Ta] N. Tanaka, “A Differential Geometric Study on Strongly Pseudo-Convex Manifolds”, Kinokuniya Co. Ltd., Tokyo, 1975. Zbl0331.53025MR399517
- [We] S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41. Zbl0379.53016MR520599
Citations in EuDML Documents
top- Nataliya Shcherbakova, Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the case
- Nataliya Shcherbakova, Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the case
- Luca Capogna, Giovanna Citti, Maria Manfredini, Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups