Constant mean curvature, flux conservation, and symmetry

As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields.In Theorem 3.5 here, we generalize that law by relaxing the topological restrictions assumed in [KKS] and by allowing a weighted mean curvature functional. We also prove a partial converse (Theorem 4.1) which roughly says that when flux is conserved along a Killing field, a hypersurface splits into two regions: one with constant (weighted) mean curvature, and one preserved by the Killing field. We demonstrate our theory by using it to derive a first integral for helicoidal surfaces of constant mean curvature in Euclidean 3-space, i.e., "twizzlers."