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Free boundary minimal annuli in geodesic balls of $\mathbb{H}^3$

Alberto Cerezo

Abstract

We construct a countable collection of one-parameter families of non-rotational minimal annuli with free boundary in geodesic balls of hyperbolic 3-space. Every surface within a given family shares a common prismatic symmetry group, and they appear as bifurcations from certain free boundary hyperbolic catenoids.

Free boundary minimal annuli in geodesic balls of $\mathbb{H}^3$

Abstract

We construct a countable collection of one-parameter families of non-rotational minimal annuli with free boundary in geodesic balls of hyperbolic 3-space. Every surface within a given family shares a common prismatic symmetry group, and they appear as bifurcations from certain free boundary hyperbolic catenoids.

Paper Structure

This paper contains 30 sections, 36 theorems, 157 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Result of the paper and discussion
  3. Strategy of the proof and difficulties
  4. Structure of the paper
  5. Open problems
  6. Acknowledgements
  7. Preliminaries
  8. A set of solutions of the system
  9. A family of minimal immersions in
  10. Geometric properties of the family of immersions
  11. Description of the symmetries of
  12. The period map
  13. Constructing minimal annuli
  14. The geometry of the immersions
  15. Catenoids in
  16. ...and 15 more sections

Key Result

Theorem 1.1

There exists an open interval $\mathcal{J} \subset \left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{3}}\right)$ satisfying the following property: let $q \in \mathcal{J} \cap \mathbb{Q}$, which we express as an irreducible fraction $q = -m/n$, $m,n \in \mathbb{N}$. Then, there exists $\varepsilon = \vareps

Figures (5)

  • Figure 5.1: Integration path $\Xi_n$.
  • Figure 5.2: The open set $\mathcal{W} \subset \mathcal{O}$ in Remark \ref{['rem:tauW']}. The boundary component of $\mathcal{W}$ in the quadrant $\{b > 1, \kappa \geq 0\}$ represents the set $\mathcal{K}$ in \ref{['eq:defK']}.
  • Figure 7.1: Geometric interpretation of the orthogonal radius. Here, ${\bf t}(u,0)$ denotes the unit tangent vector $e^{-\omega(u)}\psi_u(u,0)$.
  • Figure 8.1: Level curve $\Theta = \Theta_0$ of the period map on the plane $\{a = 1\}$ for $\Theta_0 \in \left(- \frac{1}{\sqrt{2}},-\frac{1}{\sqrt{3}}\right)$. This level set and the curve $\mu$ intersect at a point $\mu(\kappa^*) \in \mathcal{W}$ with $\kappa^* <0$.
  • Figure A.1: Integration path $\Xi_n$.

Theorems & Definitions (108)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 98 more