Journal of the Mathematical Society of Japan

Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary


Full-text: Open access


We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold $M$  endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure $\theta$  on $M$, [20]) on the total space of the canonical circle bundle $S^1 \to C(M) \stackrel{\pi}{\rightarrow} M$  (a manifold with boundary $\partial C(M) = \pi^{-1} (\partial M)$  and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface $N = \{ \varphi = 0 \} \subset \bm{H}_1$  we show that the mean curvature vector of $N \hookrightarrow \bm{H}_1$  is expressed by $H = - \frac{1}{2} \sum_{j=1}^2 X_j ( |X\varphi |^{-1} X_j\varphi ) \xi$  provided that $N$  is tangent to the characteristic direction  $T$  of $(\bm{H}_1 , \theta_0 )$, thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g.[7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion $\Psi : N \to \bm{H}_n$  of a Riemannian manifold into the Heisenberg group we show that $\Delta \Psi = 2 J T^\bot$  hence start a Weierstrass representation theory for minimal surfaces in $\bm{H}_n$.

Article information

J. Math. Soc. Japan Volume 60, Number 2 (2008), 363-396.

First available in Project Euclid: 30 May 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 32V20: Analysis on CR manifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

CR manifold with boundary minimal submanifold Fefferman metric CR Yamabe problem


DRAGOMIR, Sorin. Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary. Journal of the Mathematical Society of Japan 60 (2008), no. 2, 363--396. doi:10.2969/jmsj/06020363.

Export citation


  • N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z., 256 (2007), 661–684.
  • 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.
  • E. Barletta and S. Dragomir, On the CR structure of the tangent sphere bundle, Le Matematiche, Catania, (2)L(1995), 237–249.
  • E. Barletta, S. Dragomir and H. Urakawa, Yang-Mills fields on CR manifolds, J. Math. Phys., 47 (2006), 1–41.
  • J. K. Beem and P. E. Ehrlich, Global Lorentzian geometry, Marcel Dekker, Inc., New York-Basel, 1981.
  • 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.
  • B. Y. Chen, Geometry of submanifolds, Marcel Dekker, Inc., New York, 1973.
  • 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.
  • S. Dragomir and J. C. Wood, Sottovarietà minimali ed applicazioni armoniche, Quaderni dell'Unione Matematica Italiana, 35, Pitagora Editrice, Bologna, 1989.
  • S. Dragomir, On a conjecture of J. M. Lee, Hokkaido Math. J., 23 (1994), 35–49.
  • S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR manifolds, Progress in Mathematics, 246, Birkhäuser, Boston-Basel-Berlin, 2006.
  • J. F. Escobar, The Yamabe problem on manifolds with boundary, J. Diff. Geometry, 35 (1992), 21–84.
  • 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.
  • N. Gamara and R. Yacoub, CR Yamabe conjecture –- the conformally flat case, Pacific J. Math., 201 (2001), 121–175.
  • 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.
  • N. Garofalo and S. D. Pauls, The Bernstein problem in the Heisenberg group, preprint, 2002.
  • D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Contemp. Math., 27 (1984), 57–63.
  • D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Diff. Geometry, 25 (1987), 167–197.
  • D. Jerison and J. M. Lee, CR normal coordinates and the Yamabe problem, J. Diff. Geometry, 29 (1989), 303–344.
  • J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411–429.
  • J. M. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc., 17 (1987), 37–91.
  • 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.
  • B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459–469.
  • B. O'Neill, Semi-Riemannian geometry, Academic Press, New York-London-Paris, 1983.
  • S. D. Pauls, Minimal surfaces in the Heisenberg group, Geometriae Dedicata, 104 (2004), 201–231.
  • 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.
  • Per Tomter, Constant mean curvature surfaces in the Heisenberg group, Proc. Sympos. Pure Math., 54 (1993), 485–495.
  • S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geometry, 13 (1978), 25–41.