As first noted by Korevaar, Kusner, and Solomon, constant mean curvature implies a
homological conservation law for hypersurfaces in ambient spaces with Killing fields.
We generalize that law by relaxing the topological restrictions assumed by Korevaar
et al., and by allowing a weighted mean curvature functional. We also prove a partial
converse, 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
, i.e.,
“twizzlers”.
