# Limiting behavior of sequences of properly embedded minimal disks

## TL;DR

This paper develops a theory of minimal theta-graphs to analyze the limits of embedded minimal disks, revealing new phenomena like non-properly embedded minimal surfaces in hyperbolic space.

## Contribution

It introduces the concept of minimal theta-graphs and characterizes their limit laminations, extending the understanding of minimal surface limits in various geometries.

## Key findings

- Realization of catenoids as limit leaves without curvature blow-up
- Existence of a complete, simply connected, minimal surface in hyperbolic space that is not properly embedded
- Extension of methods to hyperbolic and more general Riemannian settings

## Abstract

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.