Journal of the Mathematical Society of Japan

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

Sorin DRAGOMIR

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Abstract

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

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

Dates
First available in Project Euclid: 30 May 2008

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1212156655

Digital Object Identifier
doi:10.2969/jmsj/06020363

Mathematical Reviews number (MathSciNet)
MR2421981

Zentralblatt MATH identifier
1148.53042

Subjects
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]

Keywords
CR manifold with boundary minimal submanifold Fefferman metric CR Yamabe problem

Citation

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. http://projecteuclid.org/euclid.jmsj/1212156655.


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