CR-Analogue of Siu-$\partial\bar{\partial}$-formula and Applications to Rigidity problem for pseudo-Hermitian harmonic maps

We give several versions of Siu's $\partial\bar{\partial}$-formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold $(M^{2m+1}, \theta)$ into a K\"ahler manifold $(N^n, g)$. We also define and study the notion of pseudo-Hermitian harmonicity for maps from $M$ into $N$. In particular, we prove a CR version of Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a K\"ahler manifold having strongly negative curvature.