Constant mean curvature, flux conservation, and symmetry

TL;DR

This paper generalizes flux conservation laws for hypersurfaces with constant mean curvature, relaxing topological constraints and introducing weighted mean curvature, with applications to helicoidal surfaces in Euclidean space.

Contribution

It extends the homological conservation law to broader settings and establishes a partial converse relating flux conservation to hypersurface splitting.

Findings

Generalized flux conservation law with weighted mean curvature

Proved a partial converse linking flux conservation to hypersurface structure

Derived a first integral for helicoidal constant mean curvature surfaces

Abstract

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."