Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary. (English) Zbl 1148.53042

The author formulates the CR Yamabe problem as the Yamabe problem for the Fefferman metric, which is reduced to finding a positive function as the solution of a nonlinear subelliptic problem of variational origin. The boundary condition of this problem turned out to be the problem of minimality of boundary in the canonical circle bundle. This condition projected on manifold is interpreted as CR Yamabe problem. In the paper, only geometric aspects are discussed.


53C40 Global submanifolds
32V20 Analysis on CR manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: DOI arXiv


[1] N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z., 256 (2007), 661-684. · Zbl 1128.49035
[2] A. Bahri and H. Brezis, Nonlinear elliptic equations, in “Topics in Geometry in memory of J. D’Atri”, (ed. S. Gindikin), Birkhäuser, Boston-Basel-Berlin, 1996, pp.,1-100.
[3] E. Barletta and S. Dragomir, On the CR structure of the tangent sphere bundle, Le Matematiche, Catania, (2)L(1995), 237-249. · Zbl 0911.32030
[4] E. Barletta, S. Dragomir and H. Urakawa, Yang-Mills fields on CR manifolds, J. Math. Phys., 47 (2006), 1-41. · Zbl 1112.53014
[5] J. K. Beem and P. E. Ehrlich, Global Lorentzian geometry, Marcel Dekker, Inc., New York-Basel, 1981. · Zbl 0462.53001
[6] I. Birindelli and E. Lanconelli, A negative answer to a one-dimensional symmetry problem in the Heisenberg group, Calc. Var. Partial Differential Equations, 18 (2003), 357-372. · Zbl 1290.35038
[7] B. Y. Chen, Geometry of submanifolds, Marcel Dekker, Inc., New York, 1973. · Zbl 0262.53036
[8] J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 129-177. · Zbl 1158.53306
[9] S. Dragomir and J. C. Wood, Sottovarietà minimali ed applicazioni armoniche, Quaderni dell’Unione Matematica Italiana, 35 , Pitagora Editrice, Bologna, 1989. · Zbl 0969.53510
[10] S. Dragomir, On a conjecture of J. M. Lee, Hokkaido Math. J., 23 (1994), 35-49. · Zbl 0797.53036
[11] S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR manifolds, Progress in Mathematics, 246 , Birkhäuser, Boston-Basel-Berlin, 2006. · Zbl 1099.32008
[12] J. F. Escobar, The Yamabe problem on manifolds with boundary, J. Diff. Geometry, 35 (1992), 21-84. · Zbl 0771.53017
[13] G. B. Folland and E. M. Stein, Estimates for the \(\overline{\partial}_{b}\)-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522. · Zbl 0293.35012
[14] N. Gamara and R. Yacoub, CR Yamabe conjecture – the conformally flat case, Pacific J. Math., 201 (2001), 121-175. · Zbl 1054.32020
[15] N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. de l’Inst. Fourier, 40 (1990), 313-356. · Zbl 0694.22003
[16] N. Garofalo and S. D. Pauls, The Bernstein problem in the Heisenberg group, preprint, 2002.
[17] D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Contemp. Math., 27 (1984), 57-63. · Zbl 0577.53035
[18] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Diff. Geometry, 25 (1987), 167-197. · Zbl 0661.32026
[19] D. Jerison and J. M. Lee, CR normal coordinates and the Yamabe problem, J. Diff. Geometry, 29 (1989), 303-344. · Zbl 0671.32016
[20] J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429. · Zbl 0595.32026
[21] J. M. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc., 17 (1987), 37-91. · Zbl 0633.53062
[22] H. Lewy, On the local character of the solution of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math., 64 (1956), 514-522. · Zbl 0074.06204
[23] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459-469. · Zbl 0145.18602
[24] B. O’Neill, Semi-Riemannian geometry, Academic Press, New York-London-Paris, 1983.
[25] S. D. Pauls, Minimal surfaces in the Heisenberg group, Geometriae Dedicata, 104 (2004), 201-231. · Zbl 1054.49029
[26] M. Ritoré and C. Rosales, Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group \(\bm{H}^n\), J. Geom. Anal., 16 (2006), 703-720. · Zbl 1129.53041
[27] Per Tomter, Constant mean curvature surfaces in the Heisenberg group, Proc. Sympos. Pure Math., 54 (1993), 485-495. · Zbl 0799.53073
[28] S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geometry, 13 (1978), 25-41. · Zbl 0379.53016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.