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

TL;DR

This paper investigates the integrability of the Hodge star mean curvature flow on 3-manifolds like 3 and 4, providing explicit solutions and Backlund transformations, advancing understanding in geometric analysis.

Contribution

It demonstrates the integrability of the flow on 3 and 4 and constructs explicit solutions and transformations, a novel contribution in geometric analysis.

Findings

Flow on 3 and 4 is integrable.

Explicit solutions to the flow are described.

Backlund transformations are established.

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.