On ${\overline{\partial }}_b$-harmonic maps from pseudo-Hermitian manifolds to Kähler manifolds

Abstract

In this paper, we consider maps from pseudo-Hermitian manifolds to Kähler manifolds and introduce partial energy functionals for these maps. First, we obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds in \cite{SSZ2013holomorphic}. Next, we investigate critical maps of the partial energy functionals, which are referred to as ${\overline{\partial }}_b$-harmonic maps and ${\partial }_b$-harmonic maps. We give a foliated result for both ${\overline{\partial }}_b$- and ${\partial }_b$-harmonic maps, generalizing a foliated result of Petit \cite{Pet2002harmonic} for harmonic maps. Then we are able to generalize Siu's holomorphicity result for harmonic maps \cite{Siu1980rigid} to the case for ${\overline{\partial }}_b$- and ${\partial }_b$-harmonic maps.

21 pages, all comments are welcome