The Willmore problem for surfaces with symmetry

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 $W=\int \mathrm{\mid }{H}_{{\mathbb{R}}^4}{\mathrm{\mid }}^2=area+\int \mathrm{\mid }{H}_{{\mathbb{S}}^3}{\mathrm{\mid }}^2$. 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 ${\tilde{G}}_{m,k}={\widehat{G}}_{m,k}\cap SO(4)$. We also describe a genus 2 example where known methods do not ensure the existence of a $W$-minimizer among surfaces with its symmetry.