Spaces of Holomorphic Immersions of Open Riemann Surfaces into the Complex Plane

Abstract

Let M be an open Riemann surface. A recent result due to Forstneriˇc and L´arusson [8] says that, for a closed conical subvariety A ⇢ Cn such that A \ {0} is an Oka manifold, the weak homotopy type of the space of non-degenerate holomorphic A-immersions of M into Cn is the same as that of the space of holomorphic (or equivalently, continuous) maps from M into A\{0}. In their paper, the authors sketch the proof of this theoremthrough claiming analogy with a related result, and invoking advanced results from complex and di↵erential geometry, including seminal theorems from Oka theory. The work contained in this thesis was motivated by the absence of a self-contained proof for the special case where A = C – as, perhaps, the first geometrically interesting case that one would consider. We remedy the absence by providing a fully detailed, self-contained proof of this case; namely, the parametric h-principle for holomorphic immersions of open Riemann surfaces into C. We outline this more precisely as follows. Take a holomorphic 1-form ✓ on M which vanishes nowhere. We denote by I(M,C) the space of holomorphic immersions of M into C, and denote by O(M,C⇤) the space of nonvanishing holomorphic functions on M. We prove, in all detail, that the continuous map I(M,C)!O(M,C⇤), f 7! df /✓, is a weak homotopy equivalence. This gives a full description of the weak homotopy type of I(M,C), as the target space O(M,C⇤) is known by algebraic topology (Remark 5.2.3).