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

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 $S^1\to C(M)\stackrel{\pi }{\to }M$ (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 $N=\{\phi =0\}\subset H_1$ we show that the mean curvature vector of $N\rightharpoonup H_1$ is expressed by $H=-\frac{1}{2}{\displaystyle {\sum }_{j=1}^2{X}_j}(|X\phi {|}^{-1}{X}_j\phi )\xi$ provided that $N$ is tangent to the characteristic direction $T$ of $(H_1,{\theta }_0)$, 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 $\Psi :N\to {\text{H}}_n$ of a Riemannian manifold into the Heisenberg group we show that $\Delta \Psi =2J{T}^{\perp }$ hence start a Weierstrass representation theory for minimal surfaces in $H_n$.