Harmonic Maps and Bi-Harmonic Maps on CR-Manifolds and Foliated Riemannian Manifolds

Abstract

This is a survey on our recent works on bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds, and also a research paper on bi-harmonic maps principal G-bundles. We will show, (1) for a complete strictly pseudoconvex CR manifolda Riemannian manifold of non-positive curvature, with finite energy and finite bienergy, must be pseudo harmonic;(2) for a smooth foliated map of a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, then it is transversally harmonic; (3) we will claim that the similar result holds for principal G-bundle over a Riemannian manifold of negative Ricci curvature.

Harmonic maps on weighted foliations

“…Let Q be the normal bundle of F and d T φ = dφ| Q , the restriction of dφ on the normal bundle Q. Then φ is said to be transversally harmonic if φ is a solution of the Eular-Largrange equation τ b (φ) = 0, where τ b (φ) = tr Q (∇ tr d T φ) is the transversal tension field of φ. Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [4,10,13,14,15,19,20,30]. However, a transversally harmonic map is not a critical point of the transversal energy functional [14]…”

“…Moreover, we study the Liouville type theorems for (F, F ′ ) f and transversally f -harmonic map, respectively. The Liouville type theorem has been studied by many researchers [11,29,34,42] on Riemannian manifolds and [10,14,15,30] on foliations. Specially, see [33,40] for f -harmonic maps on weighted Riemannian manifolds.…”

Liouville type theorem for (F;F')p-harmonic maps on foliations

“…Then φ is said to be transversally harmonic if the transversal tension field τ b (φ) = tr Q (∇ tr d T φ) vanishes, where d T φ = dφ| Q and Q is the normal bundle of F . Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [3,12,13,18]. However, a transversally harmonic map is not a critical point of the transversal energy [10]…”