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April, 2008 Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary
J. Math. Soc. Japan 60(2): 363-396 (April, 2008). DOI: 10.2969/jmsj/06020363


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 .


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Sorin DRAGOMIR. "Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary." J. Math. Soc. Japan 60 (2) 363 - 396, April, 2008.


Published: April, 2008
First available in Project Euclid: 30 May 2008

zbMATH: 1148.53042
MathSciNet: MR2421981
Digital Object Identifier: 10.2969/jmsj/06020363

Primary: 53C40
Secondary: 32V20 , 53C42

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

Rights: Copyright © 2008 Mathematical Society of Japan


Vol.60 • No. 2 • April, 2008
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