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

TL;DR

This paper investigates the types and conditions of singularities on timelike minimal surfaces in Lorentz-Minkowski 3-space, revealing unique singularity types not found in Euclidean minimal or Minkowski maximal surfaces.

Contribution

It establishes existence and non-existence theorems for singularities and provides criteria for various complex singularities specific to timelike minimal surfaces.

Findings

Various diffeomorphism types of singularities appear uniquely in timelike minimal surfaces.

Criteria for complex singularities like cuspidal butterfly and $D_4$ are established.

Duality and invariance theorems for these singularities are demonstrated.

Abstract

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.