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Surfaces via spinors and soliton equations

Iskander A. Taimanov

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.

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.
Paper Structure (4 sections, 10 theorems, 76 equations)

This paper contains 4 sections, 10 theorems, 76 equations.

Table of Contents

  1. The Weierstrass (spinor) representation of surfaces in the three-space
  2. Spectral characteristics of $D$ and conformal geo-metry of surfaces
  3. Surfaces in the four-space and the Davey--Ste-wart-son equation
  4. The Moutard transformation for the Davey--Ste-wartson II equation and its applications

Key Result

Theorem 1

For every solution $\psi$ of (diraceq) the formulae (weier) define a surface in ${\mathbb R}^3$ for which $z$ is a conformal parameter, induced metric takes the form and the potential $U$ of the Dirac operator equals to where $H$ is the mean curvature.

Theorems & Definitions (10)

  • Theorem 1: K1
  • Theorem 2: T1
  • Theorem 3: T1
  • Theorem 4: T21
  • Theorem 5: FLPP
  • Theorem 6: K2
  • Theorem 7: TDS
  • Theorem 8: MT
  • Theorem 9: T2021
  • Theorem 10: T2021