On the Willmore problem for surfaces with symmetry

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.