Star Mean Curvature Flow on 3 manifolds and its B\"acklund Transformations

The Hodge star mean curvature flow on a 3-dimensional Riemannian or pseudo-Riemannian manifold is a natural nonlinear dispersive curve flow in geometric analysis. A curve flow is integrable if the local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we show that this flow on ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$ are integrable, and describe algebraically explicit solutions to such curve flows. The Cauchy problem of the curve flows on ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$ and its B"acklund transformations follow from this construction.