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# Surveys in Differential Geometry

## Volume 19 (2014)

### Dispersive geometric curve flows

Pages: 179 – 229

DOI: https://dx.doi.org/10.4310/SDG.2014.v19.n1.a8

#### Author

#### 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 Schrödinger flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural nonlinear 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 $\mathbb{R}^3$ and on $\mathbb{R}^{2,1}$, the geometric Airy flow on $\mathbb{R}^n$, the Schrödinger 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.

#### Keywords

dispersive integrable curve flows, soliton equations

#### 2010 Mathematics Subject Classification

37K10, 37K25, 37K35, 53C44

Published 6 March 2015