Space-like Weingarten surfaces in the three-dimensional Minkowski space and their natural partial differential equations

TL;DR

The paper introduces natural principal parameters for space-like W-surfaces in Minkowski space, showing they are uniquely determined by an invariant function satisfying a specific PDE, thus solving a geometric reduction problem.

Contribution

It establishes a unique characterization of space-like W-surfaces via a natural PDE and applies this to classify linear fractional space-like W-surfaces.

Findings

Introduction of natural principal parameters for W-surfaces

Derivation of a natural non-linear PDE for the invariant function

Complete description of linear fractional space-like W-surfaces via PDEs

Abstract

On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution to the Lund-Regge reduction problem for space-like W-surfaces in Minkowski space. We apply this theory to linear fractional space-like W-surfaces and obtain the natural non-linear partial differential equations describing them. We obtain a characterization of space-like surfaces, whose curvatures satisfy a linear relation, by means of their natural partial differential equations. We obtain the ten natural PDE's describing all linear fractional space-like W-surfaces.