In this paper we show that the space of holomorphic immersions from any given open Riemann surface into the Riemann sphere is weakly homotopy equivalent to the space of continuous maps from to the complement of the zero section in the tangent bundle of . It follows in particular that this space has path components, where is the number of generators of the first homology group . We also prove a parametric version of the Mergelyan approximation theorem for maps from Riemann surfaces to an arbitrary complex manifold, a result used in the proof of our main theorem.

中文翻译：

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开放黎曼曲面浸入黎曼球体

在本文中，我们证明了从任何给定的开放黎曼曲面到黎曼球面的全纯浸入空间是弱同伦的，等价于 的切丛中从到零截面的补集的连续映射空间。特别是该空间具有路径分量，其中是第一个同源群的生成器的数量。我们还证明了从黎曼曲面到任意复流形的映射的 Mergelyan 近似定理的参数版本，该结果用于证明我们的主要定理。