The Willmore problem for surfaces with symmetry

Abstract: The Willmore Problem seeks the surface in $\mathbb{S}<sup>3\subset\mathbb{R}<sup>4$ of a given topological type minimizing the squared-mean-curvature energy $$W = \int |H_{\mathbb{R}^4}|² = area + \int |H_{\mathbb{S}<sup>3}|<sup>2$$. The longstanding Willmore Conjecture that the Clifford torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and Neves [22], but the general conjecture \cite[12] that Lawson's [18] 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 $\widehat{G}<em>{g,1}$ of $\xi</em>{g,1}$. In fact, we show each Lawson surface $\xi_{m,k}$ satisfies the analogous $W$-minimizing property under a smaller symmetry group $$\widetilde{G}<em>{m,k}=\widehat{G}</em>{m,k}\cap SO(4)$$. We also describe a genus 2 example where known methods do not ensure the existence of a $W$-minimizer among surfaces with its symmetry.