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

Abstract

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.