Minimal surfaces in pseudohermitian geometry
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, p. 129-177
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the xy-plane) in the Heisenberg group H 1 . In H 1 , identified with the euclidean space 3 , the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C 2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S 3 . This fact continues to hold when S 3 is replaced by a general pseudohermitian 3-manifold.
Classification:  35L80,  35J70,  32V20,  53A10,  49Q10
     author = {Cheng, Jih-Hsin and Hwang, Jenn-Fang and Malchiodi, Andrea and Yang, Paul},
     title = {Minimal surfaces in pseudohermitian geometry},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {1},
     year = {2005},
     pages = {129-177},
     zbl = {1158.53306},
     mrnumber = {2165405},
     language = {en},
     url = {}
Cheng, Jih-Hsin; Hwang, Jenn-Fang; Malchiodi, Andrea; Yang, Paul. Minimal surfaces in pseudohermitian geometry. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 129-177.

[B] Z. Balogh, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math. 564 (2003), 63-83. | MR 2021034 | Zbl 1051.53024

[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. | MR 1312686 | Zbl 0864.46018

[CF] P. Concus and R. Finn, On capillary free surfaces in the absence of gravity, Acta Math. 132 (1974), 177-198. | MR 670441 | Zbl 0382.76003

[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. | MR 2103983 | Zbl 1062.35046

[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. | Numdam | MR 1136846 | Zbl 0754.53013

[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 ¯ b -complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. | MR 367477 | Zbl 0293.35012

[FSS] 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

[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. | MR 1404326 | Zbl 0880.35032

[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. | MR 666108 | Zbl 0584.53021

[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. | Numdam | MR 1015799 | Zbl 0705.49022

[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. | Zbl 1193.35046

[JL1] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167-197. | MR 880182 | Zbl 0661.32026

[JL2] D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), 303-343. | MR 982177 | Zbl 0671.32016

[Jo] F. John, “Partial Differential Equations”, Springer-Verlag, 4th ed., 1982. | MR 831655 | Zbl 0472.35001

[La] H. B. Lawson Jr., Complete minimal surfaces in S 3 , Ann. of Math. 92 (1970), 335-374. | MR 270280 | Zbl 0205.52001

[Lee] J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc. 296 (1986), 411-429. | MR 837820 | Zbl 0595.32026

[LM] G. Leonardi and S. Masnou, On the isoperimetric problem in the Heisenberg group H n , to appear on Annali Mat. Pura e Appl., 2002. | MR 2177813 | Zbl pre05058523

[LR] G. Leonardi and S. Rigot, Isoperimetric sets on Carnot groups, Houston J. Math. (2003). | MR 2000099 | Zbl 1039.49037

[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. | MR 525842 | Zbl 0488.49029

[Mil] J. Milnor, “Topology from the Differentiable Viewpoint”, University of Virginia Press, 1965. | MR 226651 | Zbl 0136.20402

[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. | MR 852409

[Pan] P. Pansu, Une inegalite isoperimetrique sur le groupe de Heisenberg, C.R. Acad. Sci. Paris I 295 (1982), 127-130. | MR 676380 | Zbl 0502.53039

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

[Sp] M. Spivak, “A Comprehensive Introduction to Differential Geometry”, Vol. 3, Publish or Perish Inc., Boston, 1975. | Zbl 0306.53001

[St] S. Sternberg, “Lectures on Differential Geometry”, 2nd ed., Chelsea Publishing Company, New York, 1983. | MR 891190 | Zbl 0518.53001

[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. | MR 526976 | Zbl 0405.53045

[Ta] N. Tanaka, “A Differential Geometric Study on Strongly Pseudo-Convex Manifolds”, Kinokuniya Co. Ltd., Tokyo, 1975. | MR 399517 | Zbl 0331.53025

[We] S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25-41. | MR 520599 | Zbl 0379.53016