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We introduce a new approach to the study of timelike minimal surfaces in theLorentz-Minkowski space through a split-complex representation formula for thiskind of surface. As applications, we solve the Bj\"orling problem for timelikesurfaces and obtain interesting examples and related results. Using theBj\"orling representation, we also obtain characterizations of minimal timelikesurfaces of revolution as well as of minimal ruled timelike surfaces in theLorentz-Minkowsi space.

Bj\"orling problem for timelike surfaces in the Lorentz-Minkowski space