Surfaces via spinors and soliton equations

TL;DR

This paper surveys the spinor-based Weierstrass representation of surfaces in 3D and 4D, explores its connection to the Willmore functional, and applies it to generate novel solutions to the Davey-Stewartson II equation with unique singularity properties.

Contribution

It provides a comprehensive overview of the Weierstrass representation and introduces a new method for constructing solutions to the Davey-Stewartson II equation with specific singularity behavior.

Findings

Representation relates to the Willmore functional.

Constructed solutions have regular initial data and develop singularities.

Solutions extend smoothly outside singular points.

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.