Harmonic maps from complex Finsler manifolds

Abstract

We derive the variation formula of the∂-energy and of the ∂-energy for a smooth map from a complex Finsler manifold to an Hermitian manifold. Applying the result on a nonlinear elliptic system due to J. Jost and S. T. Yau, we obtain some existence theorems of harmonic maps from strongly Kähler Finsler manifolds to Kähler manifolds. Also, for such maps, we show that the difference between ∂-energy and∂-energy is a homotopy invariant.

A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

“…Likewise, affine harmonic maps mapping from an affine manifold into a Riemannian manifold as a new tool for studying affine structures have been introduced in [20,21]. In another direction, harmonic maps from a Finsler manifold into a Riemannian manifold have been studied in [1,9,34,35,39]. The precise definitions of these generalizations will be given in Sect.…”

Existence and Liouville theorems for V -harmonic maps from complete manifolds

“…The notion of V-harmonic maps was introduced in [5]. It includes the Hermitian harmonic maps introduced and studied in [17], the Weyl harmonic maps from a Weyl manifold into a Riemannian manifold [18], the affine harmonic maps mapping from an affine manifold into a Riemannian manifold [15], [16], and harmonic maps from a Finsler manifold into a Riemannian manifold [2], [12], [30], [33] and [31], see [5] for explanation of these relations. Another interesting special case is when V is a gradient vector field, i.e., V = ∇f for some function f : M → R, then (1.1) takes the form In this case, (1.2) is of divergence form.…”