Let M be an open Riemann surface and let Λ⊂M\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda \subset M$$\end{document} be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions M→Rn\documentclass[12pt]{minimal}
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\begin{document}$$M\rightarrow {\mathbb {R}}^n$$\end{document}, n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document}, with prescribed values on Λ\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda $$\end{document} and whose generalized Gauss map M→CPn-1\documentclass[12pt]{minimal}
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\begin{document}$$M\rightarrow \mathbb {CP}^{n-1}$$\end{document}, n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document}, avoids n hyperplanes of CPn-1\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {CP}^{n-1}$$\end{document} located in general position. In case n=3\documentclass[12pt]{minimal}
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\begin{document}$$n=3$$\end{document}, we obtain complete nonflat conformal minimal immersions whose Gauss map M→S2\documentclass[12pt]{minimal}
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\begin{document}$$M\rightarrow {\mathbb {S}}^2$$\end{document} omits two (antipodal) values of the sphere. This result is deduced as a consequence of an interpolation theorem for conformal minimal immersions M→Rn\documentclass[12pt]{minimal}
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\begin{document}$$M\rightarrow {\mathbb {R}}^n$$\end{document} into the Euclidean space Rn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^n$$\end{document}, n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document}, with n-2\documentclass[12pt]{minimal}
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\begin{document}$$n-2$$\end{document} prescribed components.
