The constrained Willmore Lawson conjecture

A constrained Willmore torus is a torus that is critical for the Willmore functional under conformal-class-preserving variations, and it is embedded when its immersion is embedded. A torus is equivariant when it is generated by an equivariant construction with respect to a one-parameter symmetry group. Constrained Willmore Lawson conjecture. Every embedded constrained Willmore torus is equivariant. The claim is known when the torus is Möbius congruent to a constant-mean-curvature torus, by results cited in the source; the general case is open and would reduce the relevant minimization questions to the equivariant setting.