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.
This work was partially supported by grants to Brian White from the Simons Foundation (#396369) and from the National Science Foundation (DMS 1404282).
"Limiting behavior of sequences of properly embedded minimal disks." J. Differential Geom. 116 (2) 281 - 319, October 2020. https://doi.org/10.4310/jdg/1603936813