Star Mean Curvature Flow on 3 manifolds and its Bäcklund Transformations

2018/01/31 by Hsiao-Fan Liu, Liu, Hsiao-Fan
Physics and Astronomy · #14H70 #37K10 #53C44 #70E40 #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cold Atom Physics and Bose-Einstein Condensates #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1801.10347

openalex publication_date 2018/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbS³ and ℍ³ are integrable, and describe algebraically explicit solutions to such curve flows. The Cauchy problem of the curve flows on \mathbbS³ and ℍ³ and its Bäcklund transformations follow from this construction.

Citations

• Bäcklund Transformations and Loop Group Actions

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