Harmonic maps from Riemann surfaces into complex Finsler manifolds

Abstract

Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the [Formula: see text]-energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the [Formula: see text]-energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kähler, we determine the second variation of the functional, and prove that any [Formula: see text]-energy minimizing harmonic map from the Riemann sphere to a weakly Kähler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.