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

令M为开放的Riemann曲面，令\（\ varLambda \ subset M \）为封闭的离散子集。在本文中，我们证明存在完全保形的最小浸入\（M \ rightarrow {\ mathbb {R}} ^ n \），\（n \ ge 3 \），并且在\（\ varLambda \）和其广义高斯映射\（M \ rightarrow \ mathbb {CP} ^ {n-1} \），\（n \ ge 3 \）避免\（\ mathbb {CP} ^ {n-1} \的n个超平面）位于一般位置。在\（n = 3 \）的情况下，我们获得其高斯图\（M \ rightarrow {\ mathbb {S}} ^ 2 \）的完全非平坦的共形最小浸入省略了球的两个（对映）值。该结果推导出作为结果的内插定理共形极小浸入的\（M \ RIGHTARROW {\ mathbb {R}} ^ N \）到欧氏空间\（{\ mathbb {R}} ^ N \） ，\ （n \ ge 3 \），并带有\（n-2 \）个规定的分量。