Pseudoharmonic maps with potential

We study pseudoharmonic maps with potential $V$, i.e. smooth maps $\phi: M \to N$ of a compact strictly psudoconvex CR manifold $M$ into a Riemannian manifold $N$ which are critical points of the functional $E_V(\phi)= \frac{1}{2} \int_M \{ \mathrm{trace}_\theta (\pi_H \phi^* h) - 2 V\circ \phi\} \theta \wedge (d \theta)^n$. We derive the first and second variation formula for $E_V$. We show that any pseudoharmonic map with a concave $C^2$ potential into a Riemannian manifold of nonpositive curvature is stable. Also, any nonconstant pseudoharmonic map $\phi : M \to S^\nu$ with a strictly convex $C^2$ potential is shown to be unstable.