Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds

We study pseudoharmonic maps from a strictly pseudoconvex CR manifold $(M, \theta)$ into a Riemannian manifold $(N, h)$, i.e. critical points of the energy functional $E(\varphi )= \frac{1}{2} \int_M \mathrm{trace}_{G_\theta} ( \pi_H \varphi^* h )\theta \wedge (d \theta)^n$. These are solutions $\varphi$ to a nonlinear subelliptic system $\Delta_b \varphi^i - h^{\alpha \bar\beta} T_\alpha (\varphi^j) T_{\bar\beta}(\varphi^k) ({\Gamma^\prime}^i_{jk} \circ \varphi ) =0$. The vertical lift (to the canonical $S^1$-bundle over $M$) of any pseudoharmonic map is harmonic with respect to the Fefferman metric. A CR map of a strictly pseudoconvex CR manifolds is pseudoharmonic if and only if it is a pseudohermitian map. We derive the second variation formula for $E(\varphi)$ and consider the corresponding notion of stability. Any nonconstant pseudoharmonic map into a negatively curved Riemannian manifold (respectively into a sphere) is stable (respectively unstable).