Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher genus Angel surfaces

Rivu Bardhan, Indranil Biswas, Shoichi Fujimori, Pradip Kumar

Abstract

We prove the existence of complete minimal surfaces in $\mathbb{R}^3$ of arbitrary genus $p\, \ge\, 1$ and least total absolute curvature with precisely two ends -- one catenoidal and one Enneper-type -- thereby solving, affirmatively, a problem posed by Fujimori and Shoda. These surfaces, which are called \emph{Angel surfaces}, generalize some examples numerically constructed earlier by Weber. The construction of these minimal surfaces involves extending the orthodisk method developed by Weber and Wolf \cite{weber2002teichmuller}. A central idea in our construction is the notion of \emph{partial symmetry}, which enables us to introduce controlled symmetry into the surface.

Higher genus Angel surfaces

Abstract

We prove the existence of complete minimal surfaces in of arbitrary genus and least total absolute curvature with precisely two ends -- one catenoidal and one Enneper-type -- thereby solving, affirmatively, a problem posed by Fujimori and Shoda. These surfaces, which are called \emph{Angel surfaces}, generalize some examples numerically constructed earlier by Weber. The construction of these minimal surfaces involves extending the orthodisk method developed by Weber and Wolf \cite{weber2002teichmuller}. A central idea in our construction is the notion of \emph{partial symmetry}, which enables us to introduce controlled symmetry into the surface.

Paper Structure

This paper contains 43 sections, 7 theorems, 116 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Overview of the construction
  3. Organization of the article
  4. Preliminaries
  5. Minimal surface in $\mathbb R^3$
  6. Weierstrass-Enneper representation
  7. Construction of minimal surfaces in ${\mathbb R}^3$
  8. The metric and ends of a minimal surface
  9. The method of Weber and Wolf to construct minimal surfaces
  10. Schwarz-Christoffel mapping
  11. Schwarz-Christoffel mapping and orthodisk weber2002teichmuller
  12. Minimal surface from a reflexive pair
  13. Homology basis of the hyperelliptic Riemann surface
  14. Generalized orthodisk and minimal surface
  15. From e-reflexive orthodisks to minimal surfaces
  16. ...and 28 more sections

Key Result

Lemma 3.4

Take an e-reflexive pair of orthodisks $(X^p_1,\,X^p_2)$ of genus $p$, and let their vertex data be $A$ and $B$ respectively. If $a_i+b_i\,\equiv\, 0\,\pmod{2}$ for every $i$, then there exists a meromorphic form $\eta_{X^p_1}$ on $R_{X^p_1}^{\mathrm{ess}}$ and a real constant $c_0$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (16)

  • Figure 1: $p=0$
  • Figure 2: $p=1$
  • Figure 3: $p=2$
  • Figure 4: $p=3$
  • Figure 5: $p=4$
  • ...and 11 more figures

Theorems & Definitions (24)

  • Definition 3.1: Generalized orthodisk
  • Definition 3.2: Enhanced conformal polygon
  • Definition 3.3: Conformal pair of e-generalized orthodisks
  • Remark 3.1
  • Remark 3.2: On notation for homology cycles
  • Definition 3.4: E-conjugate generalized orthodisks of genus $p$
  • Definition 3.5: E - reflexive generalized orthodisks of genus $p$
  • Remark 3.3
  • Lemma 3.4
  • Theorem 3.5
  • ...and 14 more