Time-like Weingarten surfaces with real principal curvatures in the three-dimensional Minkowski space and their natural partial differential equations

TL;DR

This paper investigates time-like surfaces with real principal curvatures in Minkowski space, establishing their unique determination by an invariant function satisfying a specific PDE, and applies this to linear fractional cases.

Contribution

It introduces natural principal parameters on time-like W-surfaces and characterizes these surfaces via a unique invariant function satisfying a natural PDE.

Findings

Unique determination of time-like W-surfaces by an invariant function.

Derivation of natural PDEs for linear fractional time-like W-surfaces.

Solution to the Lund-Regge reduction problem in this context.

Abstract

We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface 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 of the Lund-Regge reduction problem for time-like W-surfaces with real principal curvatures in Minkowski space. We apply this theory to linear fractional time-like W-surfaces and obtain the natural partial differential equations describing them.