Singularities of generalized timelike minimal surfaces in Lorentz-Minkowski 3-space

A timelike minimal surface in Minkowski 3-space is a surface whose induced metric is Lorentzian and with vanishing mean curvature. Such surfaces have many kinds of singularities. In this paper, we prove existence and non-existence theorems of singularities of timelike minimal surfaces, and show that various diffeomorphism types of singularities that do not appear on these Riemannian counter parts, such as minimal surfaces in Euclidean space and maximal surfaces in Minkowski space, appear on timelike minimal surfaces. We also give criteria for cuspidal butterfly, cuspidal $S_1$ singularity, $(2,5)$-cuspidal edge, cuspidal beaks and $D_4$ singularity of timelike minimal surfaces. Finally, duality and invariance theorems for these singularities and examples are given.