Harmonic Maps in Complex Finsler Geometry

Abstract

Abstract. Given a smooth map from a compact Riemann surface to a complex manifold M equipped with a strongly pseudoconvex Finsler metric F, we define a natural notion of the a-energy of the map. A harmonic map is then defined to be a critical point of the a-energy functional. Under the condition that F is weakly Kahler, we obtain the second variation formula of the functional, and prove that any a-energy minimizing harmonic map from a Riemann sphere to a weakly Kahler Finsler manifold M of positive curvature is either holomorphic or antiholomorphic. Significance of complex Finsler metrics in the study of holomorphic vector bundles, in particular its relation with the Hartshorne conjecture in complex algebraic geometry, is represented. A brief overview of complex Finsler geometry is also provided.