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 conjecture under symmetry constraints.

Contribution

It demonstrates that Lawson surfaces are W-minimizers within their symmetry groups, advancing the partial resolution of the Willmore conjecture for higher genus surfaces.

Findings

Lawson surfaces satisfy W-minimization under certain symmetries

The proof extends the Willmore conjecture to symmetric surfaces of genus g

A genus 2 example shows limitations of current methods

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 $\xi_{g,1}\subset\mathbb{S}^3$ 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 $\widehat{G}_{g,1}$ of $\xi_{g,1}$. In fact, we show each Lawson surface $\xi_{m,k}$ 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.