Contact harmonic maps

Abstract

We study contact harmonic maps, i.e. smooth maps ϕ : M → N from a strictly pseudoconvex CR manifold M into a contact Riemannian manifold N which are critical points of the functional E ( ϕ ) = 1 2 ∫ M ‖ ( d ϕ ) H , H ′ ‖ 2 θ ∧ ( d θ ) n and their generalizations. We derive the first and second variation formulae for E and study stability of contact harmonic maps. Contact harmonic maps are shown to arise as boundary values of critical points ϕ ∈ C ∞ ( Ω ¯ , N ) of the functional ∫ Ω ‖ Π H ′ ϕ ∘ ϕ ⁎ ‖ 2 d vol ( g B ) where Ω ⊂ C n + 1 is a smoothly bounded strictly pseudoconvex domain endowed with the Bergman metric g B .