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 \(W = \int |{\textbf {H}}_{\mathbb {R}^4}|^2 = \operatorname {area}+ \int H_{\mathbb {S}^3}^2\) . The longstanding Willmore Conjecture that the Clifford torus minimizes W among genus-1 surfaces is now a theorem of Marques and Neves [27], but the general conjecture [15] that Lawson’s [23] 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.
