黎曼球面上全纯等价关系的构造及其应用

Construction of Holomorphic EquivalenceRelations on Riemannian Spheres and TheirApplications

本文研究复一维连通复解析流形上的一些特殊黎曼面,包括复一维射影空间 ℂP1、 扩充复平面C∞和复球面S2。 在全纯映射和双全纯映射意义下,这三个典型的黎曼面是全纯等价。 进而在Hopf 映射下, 推出S3与ℂP1全纯等价。 基于Frankel 猜想, 讨论了复一维射影空间ℂP1到紧K¨ahler 流形上关于能量最小化的全纯映射问题。

In this paper, we study some special Riemann surfaces on complex one-dimensional connected complex analytic manifolds, including complex one-dimensional projection space ℂP1, extended complex plane C∞ and complex sphere S2. In the sense of holomorphic mapping and biholomorphic mapping, these three typical Riemann surfaces are holomorphic equivalent. Furthermore, under Hopf mapping, the holomorphic equivalence between S3and ℂP1 is derived. Based on Frankle's conjecture, the problemof holomorphic mapping of energy minimization on complex one-dimensional projective spaces ℂP1 to compact Kahler manifolds is discussed.