PARTIAL DIFFERENTIAL EQUATIONS OF TIME-LIKE WEINGARTEN SURFACES IN THE THREE-DIMENSIONAL MINKOWSKI SPACE

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 the class of linear fractional time-like W-surfaces with respect to their principal curvatures and obtain the natural partial differential equations describing them.