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The Willmore problem for surfaces with symmetry

Rob Kusner, Ying Lü, Peng Wang

TL;DR

The paper establishes that Lawson minimal surfaces $\xi_{m,k}$ minimize the Willmore energy among embedded genus-$mk$ surfaces with specified symmetry, by leveraging an orbifold framework based on the symmetry group $R_{m,k}$ and its quotients. It proves both a Willmore minimization and a uniqueness result for these symmetric competitors, extending prior results to smaller symmetry groups $\widetilde{G}_{m,k}$, $G^P_{m,k}$, and $G^Q_{m,k}$. Key techniques include a detailed analysis of the orbifold $\mathbb{S}^3/R_{m,k}$, the topology of quotient surfaces $\pi(M)$, and the Schwarz-plateau construction for the base minimal disks, combined with isotopy arguments in the equivariant category. The work also clarifies the limitations by presenting a genus-2 example where a $W$-minimizer may fail to exist under certain symmetries. Overall, the paper advances the genus-$g$ Willmore problem by tying minimization and uniqueness to explicit Lawson symmetries via orbifold geometry.

Abstract

The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding Willmore Conjecture that the Clifford torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and Neves [22], but the general conjecture [12] that Lawson's [18] minimal surface $ξ_{g,1}\subset\mathbb{S}^3$ minimizes $W$ among surfaces of genus $g>1$ remains open. Here we prove this conjecture under the additional assumption that the competitor surfaces $M\subset\mathbb{S}^3$ share the ambient symmetries $\widehat{G}_{g,1}$ of $ξ_{g,1}$. In fact, we show each Lawson surface $ξ_{m,k}$ satisfies the corresponding $W$-minimizing property under a smaller symmetry group $\widetilde{G}_{m,k}=\widehat{G}_{m,k}\cap SO(4)$. We also describe a genus 2 example where known methods do not ensure the existence of a $W$-minimizer among surfaces with its symmetry.

The Willmore problem for surfaces with symmetry

TL;DR

The paper establishes that Lawson minimal surfaces minimize the Willmore energy among embedded genus- surfaces with specified symmetry, by leveraging an orbifold framework based on the symmetry group and its quotients. It proves both a Willmore minimization and a uniqueness result for these symmetric competitors, extending prior results to smaller symmetry groups , , and . Key techniques include a detailed analysis of the orbifold , the topology of quotient surfaces , and the Schwarz-plateau construction for the base minimal disks, combined with isotopy arguments in the equivariant category. The work also clarifies the limitations by presenting a genus-2 example where a -minimizer may fail to exist under certain symmetries. Overall, the paper advances the genus- Willmore problem by tying minimization and uniqueness to explicit Lawson symmetries via orbifold geometry.

Abstract

The Willmore Problem seeks closed surfaces in of a given topological type minimizing the squared-mean-curvature energy . The longstanding Willmore Conjecture that the Clifford torus minimizes among genus- surfaces is now a theorem of Marques and Neves [22], but the general conjecture [12] that Lawson's [18] minimal surface minimizes among surfaces of genus remains open. Here we prove this conjecture under the additional assumption that the competitor surfaces share the ambient symmetries of . In fact, we show each Lawson surface satisfies the corresponding -minimizing property under a smaller symmetry group . We also describe a genus 2 example where known methods do not ensure the existence of a -minimizer among surfaces with its symmetry.

Paper Structure

This paper contains 12 sections, 9 theorems, 36 equations, 5 figures.

Table of Contents

  1. Introduction
  2. On the symmetries of the Lawson minimal surface $\xi_{m,k}$
  3. Construction of $\xi_{m,k}$
  4. Symmetries of $\xi_{m,k}$
  5. The spherical orbifold $\mathbb{S}^3/R_{m,k}$
  6. Topology of the 3-orbifold $\mathbb{S}^3/R_{m,k}$
  7. Flip-symmetries of $\mathbb{S}^3/R_{m,k}$
  8. A fundamental domain in $\mathbb{S}^3$ for $R_{m,k}$
  9. Locating $\pi(\xi_{m,k})$ and $\pi(\xi^*_{m,k})$ in $\mathbb{S}^3/R_{m,k}$
  10. Embedded ${G}_{m,k}$-symmetric surfaces projected to $\mathbb{S}^3/R_{m,k}$
  11. Uniqueness of W-minimizing embedded symmetric surfaces
  12. ${G}_{m,k}$-equivariant isotopy classes of embedded surfaces in $\mathbb{S}^3$

Key Result

Theorem 1.1

Let $\widetilde{G}_{m,k}<SO(4)$ be the halfturn-symmetry group of order $4(m+1)(k+1)$ for the embedded Lawson minimal surface $\xi_{m,k}\subset\mathbb{S}^3$ of genus $mk$, and let $G$ be any group conjugate to $\widetilde{G}_{m,k}$ in the Möbius group of conformal transformations of $\mathbb{S}^3$. The first equality holds if and only if $M$ is Möbius-congruent to $\xi_{m,k}$, and the second equa

Figures (5)

  • Figure 1: Does a $W$-minimizing sequence of genus 2 surfaces with $\widehat{K}$-symmetry converge to a $W$-minimizer of lower genus? [Image courtesy of Ken Brakke]
  • Figure 2: The fundamental domain $\mathcal{D}$ of $R_{m,k}$ and the orbifold $\mathbb{S}^3/R_{m,k}$
  • Figure 3: $\xi_{m,k}$ in $\mathcal{D}$ and $\xi_{m,k}/R_{m,k}$ in $\mathbb{S}^3/R_{m,k}$
  • Figure 4: $\xi^*_{m,k}$ in $\mathcal{D}$ and $\xi^*_{m,k}/R_{m,k}$ in $\mathbb{S}^3/R_{m,k}$
  • Figure :

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 11 more