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 $S^1 \to C(M) \stackrel{\pi}{\rightarrow} M$  (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 $\bm{H}_n$.