Journal of the Mathematical Society of Japan

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


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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 θ   on M , [20]) on the total space of the canonical circle bundle S 1 C(M) π M   (a manifold with boundary C(M)= π 1 (M)   and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface N={φ=0} H 1   we show that the mean curvature vector of N H 1   is expressed by H= 1 2 j=1 2 X j (|Xφ | 1 X j φ)ξ   provided that N   is tangent to the characteristic direction   T   of ( H 1 , θ 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 Ψ:N H n   of a Riemannian manifold into the Heisenberg group we show that ΔΨ=2J T   hence start a Weierstrass representation theory for minimal surfaces in H n .

Article information

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

First available in Project Euclid: 30 May 2008

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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. J. Math. Soc. Japan 60 (2008), no. 2, 363--396. doi:10.2969/jmsj/06020363.

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