Contact harmonic maps

We study contact harmonic maps, i.e. smooth maps $\phi: M \to N$ from a strictly pseudoconvex CR manifold $M$ into a contact Riemanian manifold $N$ which are critical points of the functional $E(\phi) = 1/2 \int_M \| (d \phi)_{H,H^\prime}\|^2 \theta \wedge (d \theta)^n$ and thenir generalizations. We derive the first and second variation formulae for $E$ and study stability of contact harmonic maps. Contact harmonic maps are shown to arise as boundary values of critical points $\phi \in C^\infty (\overline{\Omega), N)$ of the functional $\int_\Omega \| \Pi_H^\prime^\phi \circ \phi_*\|^2 d vol(g_\theta)$ where $\Omega \subset \mathbb{C}^{n+1}$ is a smoothly bounded strictly pseudoconvex domain endowed with the Bergman metric $g_B$.