Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Genus two embedded minimal surfaces in $\mathbb{S}^3$ with bidihedral symmetry

José M. Espinar, Joaquín Pérez

TL;DR

The paper proves that the Lawson genus-2 embedded minimal surface $\xi_{2,1}$ is the unique closed embedded minimal surface of genus $2$ in $\mathbb{S}^3$ whose isometry group contains the bidihedral group $D_{4h}$. The authors develop a two-stage strategy: (i) analyze a fundamental piece under $D_{4h}$ symmetry and its conjugate boundary, recoding the problem as a Plateau problem for geodesic right-angled pentagons, and (ii) solve a closing problem by balancing the length of a reflective geodesic $\gamma$ and an angle function $\Theta$ so that Schwarz reflections yield a complete, embedded, $D_{4h}$-symmetric surface. A detailed study of a two-parameter family of geodesic pentagons $\mathcal P_{l,\omega}$ (and a one-parameter family $\mathcal P_{\sigma}$) yields minimal disks $\Sigma_{l,\omega}$ and their conjugates $\Sigma^*_{l,\omega}$; the closing problem reduces to finding a unique $(l,\omega)$ with $L(l,\omega)=\pi/2$ and $\Theta(\tau)=0$, which corresponds precisely to $\xi_{2,1}$. The framework also yields a conditional uniqueness result for genus-2 symmetric surfaces and provides a pathway toward genus-$g$ extensions under similar symmetry constraints. The analysis combines Plateau theory in Meeks–Yau domains, conjugate minimal surfaces, and careful geometric control of geodesic-polygon boundaries in $\mathbb{S}^3$.

Abstract

The isometry group of the classical Lawson embedded minimal surface $ξ_{2,1}\subset \mathbb{S}^3$ of genus 2 is isomorphic to the group $O_{48}$ of isometries of a regular octahedron, of order 48. $O_{48}$ has a subgroup of index 3 isomorphic to the bidihedral group $D_{4h}=\mathbb{Z}_2\times D_4$, where $D_4$ is the dihedral group of order 8. We prove that $ξ_{2,1}$ is the unique closed embedded minimal surface of genus 2 in $\mathbb{S}^3$ whose isometry group contains $D_{4h}$.

Genus two embedded minimal surfaces in $\mathbb{S}^3$ with bidihedral symmetry

TL;DR

The paper proves that the Lawson genus-2 embedded minimal surface is the unique closed embedded minimal surface of genus in whose isometry group contains the bidihedral group . The authors develop a two-stage strategy: (i) analyze a fundamental piece under symmetry and its conjugate boundary, recoding the problem as a Plateau problem for geodesic right-angled pentagons, and (ii) solve a closing problem by balancing the length of a reflective geodesic and an angle function so that Schwarz reflections yield a complete, embedded, -symmetric surface. A detailed study of a two-parameter family of geodesic pentagons (and a one-parameter family ) yields minimal disks and their conjugates ; the closing problem reduces to finding a unique with and , which corresponds precisely to . The framework also yields a conditional uniqueness result for genus-2 symmetric surfaces and provides a pathway toward genus- extensions under similar symmetry constraints. The analysis combines Plateau theory in Meeks–Yau domains, conjugate minimal surfaces, and careful geometric control of geodesic-polygon boundaries in .

Abstract

The isometry group of the classical Lawson embedded minimal surface of genus 2 is isomorphic to the group of isometries of a regular octahedron, of order 48. has a subgroup of index 3 isomorphic to the bidihedral group , where is the dihedral group of order 8. We prove that is the unique closed embedded minimal surface of genus 2 in whose isometry group contains .

Paper Structure

This paper contains 51 sections, 39 theorems, 221 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The standard model of $\mathbb{S}^3$
  4. The conformal model of $\mathbb{S}^3\setminus\{ -{\bf k}\}$
  5. Rotation Killing fields
  6. Minimal surfaces in $\mathbb{S} ^3$
  7. Minimal helicoids
  8. The Lawson minimal surface of genus 2
  9. Conjugation of minimal surfaces in $\mathbb{S}^3$
  10. The bidihedral group $D_{4h}$ as a subgroup of $O(4)$
  11. The main theorem and outline of its proof
  12. The fundamental piece of a $D_{4h}$-symmetric minimal surface of genus $2$
  13. The Lawson surface $\xi_{2,1}$ viewed as a $D_{4h}$-invariant surface
  14. Conjugate of a fundamental piece $\mathcal{F}^*$
  15. The geodesic pentagons
  16. ...and 36 more sections

Key Result

Theorem 2.2

Given a Jordan curve $\mathcal{P}$ in the boundary of a Meeks-Yau type domain $U$, there exists a branched minimal immersion $f\colon \overline{\hbox{D}}\to U$ with boundary $f(\partial \hbox{D})=\mathcal{P}$, which is smooth in $\hbox{D}$ and has minimal area among all such maps. Furthermore:

Figures (17)

  • Figure 1: In the conformal model, the intersection of $\overline{\hbox{B}_4^+}$ with the totally geodesic two-spheres $\mathcal{S}_1=\mathbb{S}^3\cap \langle {\bf e}\rangle ^{\perp}$, $\mathcal{S}_2=\mathbb{S}^3\cap \langle {\bf i}\rangle ^{\perp}$, $\mathcal{S}_3 =\mathbb{S}^3\cap \langle {\bf j}\rangle ^{\perp}$. The boundary of $\overline{\hbox{B}_4^+}$ is $\mathcal{S}_4$.
  • Figure 2: The gray disk $\mathcal{D}_{\mathcal{L}}$ produces the Lawson surface $\xi_{2,1}$ after Schwarz reflection in the blue geodesic arcs. The red arcs are geodesics in $\mathcal{D}_{\mathcal{L}}$ of reflective symmetry.
  • Figure 3: Every embedded, $D_{4h}$-invariant minimal surface of genus $2$ is made of 16 copies of the disk $\mathcal{F}^*$ shaded in gray, which is described in Proposition \ref{['lem3.2']}.
  • Figure 4: The subdisk $\mathcal{T}$, shown in orange, represents one quarter of $\mathcal{D}_{\mathcal{L}}$. The mirror image $\mathcal{R}^{\perp}(\mathcal{T})$ across $\mathcal{S}(\Gamma^{\perp})$ is shown in gray, while the $\pi$-rotation $\mathcal{R}^*(\mathcal{T})$ about $[{\bf p}_1,{\bf q}_1]$ appears in purple.
  • Figure 5: Top left: The pentagon $\mathcal{P} _{l,\omega}$ for the choice of parameters $(l,\omega)=(\pi/3,\pi/4)$ with its edges $\delta_+,\delta_-$ in green, $\beta _+,\beta _-$ in blue, and $\alpha$ in red. Top center:$\mathcal{P} _{l,\omega}$ for $(l,\omega)=(\pi/3,0)$, contained in $\mathcal{S}_3$. Top right:$\mathcal{P} _{l,\omega}$ for $(l,\omega)=(\pi/2,\pi/4)$, contained in $\mathcal{S}_4$. Bottom left:$\mathcal{P} _{l,\omega}$ for $(l,\omega)=(2\pi/3,-\pi/4)$, a choice of parameters in $\mathfrak{C}_2$. Bottom right: The pentagon $\mathcal{P}_{\sigma}$ for the value $\sigma=\pi/4$.
  • ...and 12 more figures

Theorems & Definitions (82)

  • Conjecture 1.1
  • Definition 2.1
  • Theorem 2.2: Meeks-Yau my2
  • Proposition 2.3: Lawson la3, Proposition 13.1
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 72 more