On the Willmore problem for surfaces with symmetry

Abstract: The Willmore Problem seeks the surface in $\mathbb S^{3\subset\mathbb} R<sup>4$ of a given topological type minimizing the squared-mean-curvature energy $$W = \int |\mathbf{H}<em>{\mathbb{R}^4}|² = \operatorname{area} + \int H</em>{\mathbb{S}<sup>3}<sup>2$$. 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 $\xi_{g,1}\subset\mathbb S<sup>3$ minimizes $W$ among surfaces of genus $g&gt;1$ remains open. Here we prove this conjecture under the additional assumption that the competitor surfaces $M\subset\mathbb S<sup>3$ 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 ${G}<em>{m,k}&lt;SO(4)$, using a local computation of the orbifold Euler number $\chi_o(M/{G}</em>{m,k})$ 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.