The star mean curvature flow on 3-sphere and hyperbolic 3-space

The Hodge star mean curvature flow on a 3‑dimensional Riemannian or pseudo-Riemannian manifold is one of nonlinear dispersive curve flows in geometric analysis. Such a curve flow is integrable as its local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we show that these flows on a 3‑sphere and 3‑dimensional hyperbolic space are integrable, and describe algebraically explicit solutions to such curve flows. Solutions to the (periodic) Cauchy problems of such curve flows on a 3‑sphere and 3‑dimensional hyperbolic space and its Bäcklund transformations follow from this construction.

Keywords

moving frames, Hodge star MCF, Gross–Pitaevskii equation, periodic Cauchy problems, Bäcklund transformation

2010 Mathematics Subject Classification

14H70, 37K10, 53C44, 70E40

Full Text (PDF format)

Received 28 March 2019

Accepted 20 September 2019

Published 9 October 2020