2016/10/04 by Yuxin Dong, Dong, Yuxin
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1610.01032
openalex publication_date 2016/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate critical maps of the horizontal energy functional E_H,\widetildeH(f) for maps between two pseudo-Hermitian manifolds (M2m+1,H(M),J,θ) and (N2n+1,\widetildeH(N), \widetildeJ,\widetildeθ). These critical maps are referred to as (H,\widetildeH)-harmonic maps. We derive a CR Bochner formula for the horizontal energy density |df_H, \widetildeH|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,\widetildeH)-harmonic maps. We also introduce (H,\widetildeH)-pluriharmonic, (H,\widetildeH)-holomorphic maps between these manifolds, which provide us examples of (H,\widetildeH)-harmonic maps. Moreover, a Lichnerowicz type result is established to show that foliated (H,\widetilde H)-holomorphic maps are actually minimizers of E_H,\widetildeH(f) in their foliated homotopy classes. We also prove some unique continuation results for characterizing either horizontally constant maps or foliated (H,\widetildeH)-holomorphic maps. Furthermore, Eells-Sampson type existence results for (H,\widetildeH)-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,\widetildeH)-harmonic maps and Siu type strong rigidity results for compact regular Sasakian manifolds with either strongly negative horizontal curvature or adequately negative horizontal curvature.