We develop a theory of "minimal $\theta$-graphs" and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of embedded minimal disks, even when there is no curvature blow-up. Our methods work in a more general Riemannian setting, including hyperbolic space. This allows us to establish the existence of a complete, simply connected, minimal surface in hyperbolic space that is not properly embedded.

中文翻译：

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限制正确嵌入的最小磁盘序列的行为

我们开发了“最小$\theta$-graphs”理论并描述了此类表面的极限叠层的行为，包括对其极限叶和曲率膨胀集的理解。我们用它来证明，即使没有曲率膨胀，也可以将欧几里得空间中的悬链线族实现为嵌入的最小圆盘序列的极限叶。我们的方法适用于更一般的黎曼设置，包括双曲空间。这使我们能够在未正确嵌入的双曲空间中建立完整的、简单连接的、最小表面的存在。