On $(H,\tilde{H})$-harmonic Maps between pseudo-Hermitian manifolds

In this paper, we investigate critical maps of the horizontal energy functional $E_{H,\tilde{H}}(f)$ for maps between two pseudo-Hermitian manifolds $(M^{2m+1},H(M),J,\theta )$ and $(N^{2n+1},\widetilde{H}(N), \widetilde{J},\widetilde{\theta})$\mathrm{.}Thesecriticalmapsarereferredtoas$(H,\widetilde{H})$-harmonic maps. We derive a CR Bochner formula for the horizontal energy density $\mathrm{\mid }d{f}_{H,\tilde{H}}{\mathrm{\mid }}^2$, and introduce a Paneitz type operator acting on maps to refine the Bochner formula. As a result, we are able to establish some Bochner type theorems for $(H,\tilde{H})$-harmonic maps. We also introduce $(H,\tilde{H})$-pluriharmonic, $(H,\tilde{H})$-holomorphic maps between these manifolds, which provide us examples of $(H,\tilde{H})$-harmonic maps. Moreover, a Lichnerowicz type result is established to show that foliated $(H,\tilde{H})$-holomorphic maps are actually minimizers of $E_{H,\tilde{H}}(f)$ in their foliated homotopy classes. We also prove some unique continuation results for characterizing either horizontally constant maps or foliated $(H,\tilde{H})$-holomorphic maps. Furthermore, Eells-Sampson type existence results for $(H,\tilde{H})$-harmonic maps are established if both manifolds are compact Sasakian and the target is regular with non-positive horizontal sectional curvature. Finally, we give a foliated rigidity result for $(H,\tilde{H})$-harmonic maps and Siu type strong rigidity results for compact regular Sasakian manifolds with either strongly negative horizontal curvature or adequately negative horizontal curvature.