Harmonic Maps in Complex Finsler Geometry
Harmonic Maps in Complex Finsler Geometry
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DOI：
10.1007/978-3-0348-7968-2_8
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发表时间：
2004
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影响因子：
--
通讯作者：
Seiki Nishikawa
中科院分区：
文献类型：
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作者：
Seiki Nishikawa
给定一个从紧致Riemann曲面到复流形M的光滑映射，该复流形M具有强伪凸Finsler度量F，我们定义了该映射的a-能量的自然概念.然后定义调和映射为a-能量泛函的临界点。在F是弱Kahler流形的条件下，得到了该泛函的第二变分公式，并证明了从Riemann球面到正曲率弱Kahler Finsler流形M的任何α-能量极小调和映射是全纯的或反全纯的.阐述了复Finsler度量在全纯向量丛研究中的重要意义，特别是它与复代数几何中的哈茨霍恩猜想的关系.还提供了复杂的芬斯勒几何的简要概述。
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.