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Limiting behavior of sequences of properly embedded minimal disks

David Hoffman, Brian White

Abstract

We develop a theory of "minimal $θ$-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.

Limiting behavior of sequences of properly embedded minimal disks

Abstract

We develop a theory of "minimal -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.

Paper Structure

This paper contains 17 sections, 36 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The mathematical advances in this paper
  3. An outline of the sections of this paper
  4. $\theta$-graphs
  5. The relationship between spanning $\theta$-graphs and the surfaces of Theorem \ref{['qualExistence']}
  6. $\theta$-graphs considered as graphs in the simply connected covering of $W\setminus Z$
  7. Properties of limit laminations of sequences of minimal spanning $\theta$-graphs
  8. Existence of minimal $\theta$-graphs with prescribed boundary
  9. Smooth convergence at the boundary
  10. Necessary conditions for a lamination to appear as the limit leaves of the limit lamination of a sequence of minimal $\theta$-graphs.
  11. Specifying the rotationally invariant leaves of a limit lamination
  12. The hyperbolic case I. Existence of $\theta$-graphs with prescribed boundary at infinity: Theorem \ref{['UniqueEmbeddedDisk']} in the hyperbolic case
  13. The Hyperbolic case II. Necessary Conditions for a lamination to appear as limit leaves of a limit lamination: Section \ref{['necessaryconditions']} in the hyperbolic case.
  14. The Hyperbolic case III. Specifying the rotationally invariant leaves of a limit lamination: Section \ref{['SpecifyingLeaves']} in the hyperbolic case.
  15. The divergence theorem
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathbf{B}\subset \mathbf{R} ^3$ be the unit ball and let $Z\subset \mathbf{R} ^3$ be the vertical coordinate axis. Suppose $D_n$ is a sequence of properly embedded minimal disks in the ball $\mathbf{B}$ with the property that each disk $D=D_n$ satisfies Then there is a subsequence $D_{n(i)}$, a relatively closed subset $K$ of $\mathbf{B}\cap Z$, and a minimal lamination $\mathcal{L}$ of $\m

Figures (1)

  • Figure 1: Limit Leaves $\mathcal{M}$ in the unit ball $\mathbf{B}$. Depicted here in cross section is the lamination $\mathcal{M}$ of $\mathbf{B}$ consisting of all area-minimizing catenoids with axis $Z$ and symmetry plane $\{z=0\}$, together with all horizontal disks that are disjoint from the catenoids. Essentially any symmetric sublamination $\mathcal{M}^*$ of $\mathcal{M}$ can be realized as the set of limit leaves of a limit lamination of a sequence of properly embedded minimal disks in $\mathbf{B}$. This is proved in Theorem \ref{['realizing-M(T)']}.

Theorems & Definitions (73)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2: Simple examples of $\theta$-graphs
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.7: Boundary regularity theorem for minimal $\theta$-graphs
  • Theorem 2.8
  • Theorem 2.10
  • ...and 63 more