Surveys in Differential Geometry

Volume 19 (2014)

Dispersive geometric curve flows

Pages: 179 – 229

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

Author

Chuu-Lian Terng (Department of Mathematics, University of California at Irvine)

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

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