Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dihedralization of Minimal Surfaces in $\mathbb{R}^3$

Ramazan Yol

Abstract

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes with varying angle. We will study the limit of such surfaces when the angle converges to 0. In many cases, these limits are simpler than the original surface, and can be used in conjunction with the implicit function theorem to give new existence proofs of the original surfaces with small dihedral angle. This approach has led to the discovery of new minimal surfaces as well.

Dihedralization of Minimal Surfaces in $\mathbb{R}^3$

Abstract

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes with varying angle. We will study the limit of such surfaces when the angle converges to 0. In many cases, these limits are simpler than the original surface, and can be used in conjunction with the implicit function theorem to give new existence proofs of the original surfaces with small dihedral angle. This approach has led to the discovery of new minimal surfaces as well.
Paper Structure (16 sections, 15 theorems, 52 equations, 19 figures)

This paper contains 16 sections, 15 theorems, 52 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Geometry of the Weierstrass Representation
  3. The Chen-Gackstatter Surface of genus $3(n-1)$ with dihedral symmetry
  4. Period Problem
  5. Extension and Types of Ends
  6. Dihedralized Weber Wolf Surface
  7. Period Problem
  8. Extension, Types of Ends and Growth rates
  9. Dihedralized Catenoidal Costa-Wohlgemuth Surface
  10. Period Problem
  11. Extension, Types of Ends and Growth rates
  12. Dihedralized Singly Period Karcher Scherk Surface with Handles
  13. Period Problem of $DKS_n$
  14. Reparametrization of the Limit
  15. Solution to the Period Problem
  16. ...and 1 more sections

Key Result

Theorem 1.3

For sufficiently large $n\in \mathbb{N}$, there exists an $n$-dihedral, finite type minimal surface $DCCW_{n}$ of genus $2n-2$ with four catenoidal ends, which is invariant under a reflection with respect to a horizontal plane. Moreover, as $n \rightarrow \infty$, $DCCW_{n}$ converges to an embedded

Figures (19)

  • Figure 1: Dihedralization of Scherk Tower
  • Figure 2: $DCCW_n$ and its limit $\mathbb{R}^3$
  • Figure 3: DKS$_n$ and its limit $\mathbb{R}^3$
  • Figure 4: Dihedralized Chen-Gackstatter Surface, $\alpha=1/5$
  • Figure 5: A minimal octagon in $\mathbb{R}^3$ corresponding to $DE_{3,n}$
  • ...and 14 more figures

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['DEtheorem']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['DEoctagon']}
  • ...and 23 more