On the Willmore problem for surfaces with symmetry

TL;DR

This paper proves that Lawson's minimal surfaces minimize the Willmore energy among symmetric surfaces of the same genus, extending the understanding of the Willmore problem under symmetry constraints.

Contribution

It demonstrates that Lawson surfaces are energy minimizers within their symmetry class, using orbifold Euler number computations to exclude non-minimizing configurations.

Findings

Lawson surfaces satisfy the Willmore minimization under certain symmetries.

Orbifold Euler number calculations help exclude non-minimizing intersection patterns.

A genus 2 example shows potential non-existence of solutions with certain symmetries.

Abstract

The Willmore Problem seeks the surface in $\mathbb S^3\subset\mathbb R^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |\mathbf{H}_{\mathbb{R}^4}|^2 = \operatorname{area} + \int H_{\mathbb{S}^3}^2$. The longstanding Willmore Conjecture that the Clifford torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and Neves [19], but the general conjecture [10] that Lawson's [16] minimal surface $\xi_{g,1}\subset\mathbb S^3$ 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 of $\xi_{g,1}$. Specifcally, we show each Lawson surface $\xi_{m,k}$ satisfies the analogous $W$-minimizing property under a somewhat smaller symmetry group ${G}_{m,k}<SO(4)$, using a local computation of the orbifold Euler number $\chi_o(M/{G}_{m,k})$ to exclude certain intersection patterns of $M$ with the great circles fixed by generators of ${G}_{m,k}$. We also describe a genus 2 example where the Willmore Problem may not be solvable among surfaces with its symmetry.