Geodesic
Immersions of open Riemann~surfaces into~the~Riemann sphere
Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 562-581.

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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 $\mathbb{CP}^1$. It follows in particular that this space has $2^k$ path components, where $k$ is the number of generators of the first homology group $H_1(M,\mathbb{Z})=\mathbb{Z}^k$. 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.
Keywords: Riemann surface, holomorphic immersion, meromorphic function, $\mathrm{h}$-principle, weak homotopy equivalence. 
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F. Forstnerič. Immersions of  open Riemann~surfaces into~the~Riemann sphere. Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 562-581. http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a15/

[1] A. Alarcón, F. Forstnerič, “Null curves and directed immersions of open Riemann surfaces”, Invent. Math., 196:3 (2014), 733–771 | DOI | MR | Zbl

[2] H. Behnke, K. Stein, “Entwicklung analytischer Funktionen auf Riemannschen Flächen”, Math. Ann., 120 (1947), 430–461 | DOI | MR | Zbl

[3] A. Boivin, B. Jiang, “Uniform approximation by meromorphic functions on Riemann surfaces”, J. Anal. Math., 93 (2004), 199–214 | DOI | MR | Zbl

[4] K. Cieliebak, Y. Eliashberg, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, Amer. Math. Soc. Colloq. Publ., 59, Amer. Math. Soc., Providence, RI, 2012, xii+364 pp. | DOI | MR | Zbl

[5] J. E. Fornæss, F. Forstnerič, E. F. Wold, “Holomorphic approximation: the legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan”, Advancements in complex analysis. From theory to practice, Springer, Cham, 2020, 133–192 ; arXiv: 1802.03924 | DOI | Zbl

[6] F. Forstnerič, “Noncritical holomorphic functions on Stein manifolds”, Acta Math., 191:2 (2003), 143–189 | DOI | MR | Zbl

[7] F. Forstnerič, Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Ergeb. Math. Grenzgeb. (3), 56, 2nd ed., Springer, Cham, 2017, xiv+562 pp. | DOI | MR | Zbl

[8] F. Forstnerič, “Mergelyan's and Arakelian's theorems for manifold-valued maps”, Mosc. Math. J., 19:3 (2019), 465–484 | DOI | MR | Zbl

[9] F. Forstnerič, F. Lárusson, “The parametric $h$-principle for minimal surfaces in $\mathbb{R}^n$ and null curves in $\mathbb{C}^n$”, Comm. Anal. Geom., 27:1 (2019), 1–45 | DOI | MR | Zbl

[10] F. Forstnerič, M. Slapar, “Stein structures and holomorphic mappings”, Math. Z., 256:3 (2007), 615–646 | DOI | MR | Zbl

[11] T. W. Gamelin, Uniform algebras, 2nd ed., Chelsea Publishing Co., New York, 1984, xiii+257 pp. | MR | Zbl | Zbl

[12] M. Gromov, Partial differential relations, Ergeb. Math. Grenzgeb. (3), 9, Springer-Verlag, Berlin, 1986, x+363 pp. | DOI | MR | MR | Zbl

[13] M. L. Gromov, “Convex integration of differential relations. I”, Math. USSR-Izv., 7:2 (1973), 329–343 | DOI | MR | Zbl

[14] M. L. Gromov, Ya. M. Éliashberg, “Nonsingular mappings of Stein manifolds”, Funct. Anal. Appl., 5:2 (1971), 156–157 | DOI | MR | Zbl

[15] R. C. Gunning, R. Narasimhan, “Immersion of open Riemann surfaces”, Math. Ann., 174 (1967), 103–108 | DOI | MR | Zbl

[16] M. W. Hirsch, “Immersions of manifolds”, Trans. Amer. Math. Soc., 93:2 (1959), 242–276 | DOI | MR | Zbl

[17] D. Kolarič, “Parametric H-principle for holomorphic immersions with approximation”, Differential Geom. Appl., 29:3 (2011), 292–298 | DOI | MR | Zbl

[18] E. Michael, “Continuous selections. I”, Ann. of Math. (2), 63:2 (1956), 361–382 | DOI | MR | Zbl

[19] E. A. Poletsky, “Stein neighborhoods of graphs of holomorphic mappings”, J. Reine Angew. Math., 2013:684 (2013), 187–198 | DOI | MR | Zbl

[20] C. Runge, “Zur Theorie der Eindeutigen Analytischen Functionen”, Acta Math., 6:1 (1885), 229–244 | DOI | MR | Zbl

[21] S. Smale, “The classification of immersions of spheres in Euclidean spaces”, Ann. of Math. (2), 69:2 (1959), 327–344 | DOI | MR | Zbl

[22] A.Ġ. Vitushkin, “Necessary and sufficient conditions on a set in order that any continuous function analytic at the interior points of the set may admit of uniform approximation by rational fractions”, Soviet Math. Dokl., 7 (1966), 1622–1625 | MR | Zbl

[23] A. G. Vitushkin, “The analytic capacity of sets in problems of approximation theory”, Russian Math. Surveys, 22:6 (1967), 139–200 | DOI | MR | Zbl