Starting from the pioneering work by Meeks, complete nonorientable minimal surfaces with finite total curvature have been studied by many researchers. However, it seems that there are no known examples all of whose ends are embedded except for Kusner’s flat-ended N-noids. In this paper, we show the existence of a 1-parameter family of complete \({\mathbf{Z}}_N\)-invariant conformal minimal immersions from finitely punctured real projective planes into \({\mathbf{R}}^3\), each of which has \(N+1\) catenoidal ends, for any odd integer \(N\ge 3\). This family gives a deformation from an \((N+1)\)-noid with N catenoidal ends and a planar end to Kusner’s flat-ended N-noid. We also give a nonexistence result for such surfaces for any even integer \(N\ge 2\).

从米克斯的开创性工作开始，许多研究人员已经研究了具有有限总曲率的完全不可定向极小曲面。然而，似乎除了 Kusner 的平端N -noids之外，似乎没有已知的所有末端都嵌入的例子。在本文中，我们展示了完整的\({\mathbf{Z}}_N\) -不变共形最小浸没从有限穿孔的实射影平面到\({\mathbf{R}}^ 3\)，每个都有\(N+1\)悬链线末端，对于任何奇数整数\(N\ge 3\)。这个族给出了从\((N+1)\) -noid 与N的变形悬链线末端和 Kusner 平端N -noid 的平面末端。对于任何偶数\(N\ge 2\) ，我们还给出了此类曲面的不存在结果。