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

摘要: 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。

Abstract: 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 problem of holomorphic mapping of energy minimization on complex one-dimensional projective spaces ℂP¹ to compact Kahler manifolds is discussed.