Dispersive Geometric Curve Flows

TL;DR

This paper surveys various integrable dispersive geometric curve flows, their construction from soliton equations, and discusses their Hamiltonian structure and initial value problems.

Contribution

It provides a systematic method to derive integrable curve flows from Lax pairs and explores their geometric and Hamiltonian properties.

Findings

Identification of several integrable geometric curve flows.

Method to construct these flows from Lax pairs of soliton equations.

Discussion of Hamiltonian structures and initial value problems.

Abstract

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flow is integrable if the evolution equation of the local differential invariants of a solution of the curve flow is a soliton equation. For example, the Hodge star mean curvature flow on $R^3$ and on $R^{2,1}$, the geometric Airy flow on $R^n$, the Schrodingier flow on compact Hermitian symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a compact Lie group are integrable. In this paper, we give a survey of these results, describe a systematic method to construct integrable curve flows from Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the Cauchy problem of these curve flows.