Surfaces via spinors and soliton equations

Abstract

This article surveys the Weierstrass representation of surfaces in the three- and four-dimensional spaces, with an emphasis on its relation to the Willmore functional. We also describe an application of this representation to constructing a new type of solutions to the Davey-Stewartson II equation. They have regular initial data, gain one-point singularities at certain moments of time, and extend to smooth solutions for the remaining times.

arXiv comment18 pages

References in corpus (8)

• Surfaces in the four-space and the Davey--Stewartson equations
• Two-dimensional Dirac operator and surface theory
• The Weierstrass representation of closed surfaces in $R^3$
• The Moutard transformation for the Davey-Stewartson II equation and its geometrical meaning
• Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy
• The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres
• The Moutard transformation of two-dimensional Dirac operators and conformal geometry of surfaces in the four-space
• The Weierstrass representation of discrete isotropic surfaces in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$

Cited by in corpus (1)

• Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution