Immersions of open Riemann surfaces into the Riemann sphere

Abstract

In this paper we show that the space of holomorphic immersions from any given open Riemann surface, $M$, into the Riemann sphere ${\mathbb{CP}}^1$ is weakly homotopy equivalent to the space of continuous maps from $M$ to the complement of the zero section in the tangent bundle of ${\mathbb{CP}}^1$. We show in particular that this space has $2^k$ path components, where $H_1(M,\mathbb{Z})={\mathbb{Z}}^k$. We also prove a parametric version of Mergelyan approximation theorem for maps from Riemann surfaces into any complex manifold, a result used in the proof of our main theorem.

Izvestiya: Math. 85:3 (2021), to appear

References in corpus (1)

• Some connections of complex dynamics

Cited by in corpus (1)

• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$