Conformal deformation of spacelike surfaces in Minkowski space

TL;DR

This paper studies how spacelike surfaces in Minkowski space can be conformally deformed at second order, revealing that isothermic surfaces are special singular solutions and that most deformable surfaces are not isothermic.

Contribution

It constructs an exterior differential system for conformal deformations of spacelike surfaces and identifies isothermic surfaces as singular solutions, highlighting differences from 3D conformal geometry.

Findings

Isothermic surfaces are singular solutions of the deformation system.

Most second order deformable surfaces are not isothermic.

The deformation system differs from the 3D case.

Abstract

We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and singular solutions. In particular, we show that isothermic surfaces are singular solutions of the system, which implies that a generic second order deformable surface is not isothermic. This differs from the situation in 3-dimensional conformal geometry, where isothermic surfaces coincide with deformable surfaces.