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

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 $S^1 \to C(M) \to M$ (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 $N= \{\varphi = 0\} \subset \mathbf{H}_1$ we show that the mean curvature vector of $N \hookrightarrow \mathbf{H}_1$ 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 $(\mathbf{H}_1 , \theta_0)$, 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 $\Psi : N \to \mathbf{H}_n$ 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 $\mathbf{H}_n$.