Oka-1 manifolds Alarcón López, Antonio Forstnerič, Franc Riemann surface Complex curve Oka-1 manifold Oka manifold K3 surface Alarcón is partially supported by the State Research Agency (AEI) via the grants PID2020-117868GB-I00 and PID2023-150727NB-I00, and the “Maria de Maeztu” Unit of Excellence IMAG, reference CEX2020-001105-M, funded by MICIU/AEI/10.13039/501100011033/, and the Junta de Andalucía grant P18-FR-4049, Spain. Forstnerič is supported by the European Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291 and N1-0237 from ARIS, Republic of Slovenia. In this paper we begin a systematic study of the class of complex manifolds which are universal targets of holomorphic maps from open Riemann surfaces. We call them Oka-1 manifolds, by analogy with Oka manifolds that are universal targets of holomorphic maps from Stein manifolds of arbitrary dimension. We prove that every complex manifold which is dominable at most points by spanning tubes of complex lines in affine spaces is an Oka-1 manifold. In particular, a manifold dominable by C^n at most points is an Oka-1 manifold. We provide many examples of Oka-1 manifolds among compact complex surfaces, including all Kummer surfaces and all elliptic K3 surfaces. We show that the class of Oka-1 manifolds is invariant under Oka-1 maps inducing a surjective homomorphism of fundamental groups; this includes holomorphic fibre bundles with connected Oka fibres. In another direction, we prove that every bordered Riemann surface admits a holomorphic map with dense image in any connected complex manifold. The analogous result is shown for holomorphic Legendrian immersions in an arbitrary connected complex contact manifold. 2025-09-03T10:16:39Z 2025-09-03T10:16:39Z 2025-08-21 journal article Alarcón, A., Forstnerič, F. Oka-1 manifolds. Math. Z. 311, 33 (2025). https://doi.org/10.1007/s00209-025-03833-4 https://hdl.handle.net/10481/106034 10.1007/s00209-025-03833-4 eng http://creativecommons.org/licenses/by-nc-nd/3.0/ open access Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License Springer Nature