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1) We will always assume our pseudohermitian structure is oriented and h1¯1 = 1. h. h. h. h. 4) ω1 1 + ω¯1 1 = 0. 3) is called the (pseudohermitian) torsion. Since h1¯1 = 1, A¯1¯1 = h1¯1 A1 ¯1 = A1 ¯1 . And A11 is just the complex conjugate of A¯1¯1 . 5) ¯ dω1 1 = W θ1 ∧θ1 + 2iIm(A11,¯1 θ1 ∧Θ) where W is the Tanaka-Webster curvature. 4). This ω is just the one used in previous sections. Write Z1 = 21 (e1 − ie2 ) for real vectors e1 , e2 . It follows that e2 = Je1 . Let e1 = Re(θ1 ), e2 = Im(θ1 ).

Math. , 296 (1986) 411-429. [La] Lawson, H. , Complete minimal surfaces in S 3 , Ann. , 92 (1970) 335-374. [Mik] Miklyukov, V. , On a new approach to Bernstein’s theorem and related questions for equations of minimal surface type, Mat. , 108(150) (1979) 268-289; English transl. in Math. , 36 (1980) 251-271. , Topology from the Differentiable Viewpoint, University of Virginia Press, 1965. , Application de l’Analyse ` a la G´ eom´ etrie, Paris, Bachelier, 1850. , New York, 1986. , M´ etriques de Carnot-Carath´ eodory et quasiisom´ etries des espaces sym´ etriques de rang un, Ann.

11) holds for α∗ p ∈ SF (u). Let NF∗ (u) ≡ |α ∗ | for p ∈ Ω\SF (u). 10 ) NF∗ (v)v = F ∗ · NF (v). 10 ) that NF∗ (u)(u − v) = 0. So NF∗ (u) is tangent to Γ (at p). 11). 3. 11) for all p ∈ Γ . Let dA denote the volume element of Γ , induced from R2m . 11)) 0= Γ div(α∗ ) d(volume) (by the divergence theorem) = Ω div F ∗ d(volume) > 0 = Ω by assumption. We have reached a contradiction. D. , xm , ym ). 12) geometrically: (dxj ∧ dyj means deleting dxj ∧ dyj ) 30 α∗ · νdA = Σj=m j=1 [(uyj + xj )dyj + (uxj − yj )dxj ]∧ dx1 ∧ dy1 ∧ ...

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