Geometry of Timelike Minimal Surfaces and Null Curves

Abstract

In this chapter, we investigate the behavior of the Gaussian curvature of timelike minimal surfaces with or without singular points in the 3-dimensional Lorentz–Minkowski space. For timelike minimal surfaces without singular points, we prove that the sign of the Gaussian curvature, which corresponds to diagonalizability of the shape operator, of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also determine the behavior of the Gaussian curvature near cuspidal edges, swallowtails, and cuspidal cross caps on timelike minimal surfaces. We show that there are no umbilic points near cuspidal edges on a timelike minimal surface. Near swallowtails, we show that the sign of the Gaussian curvature is negative, that is, we can take always real principal curvatures near swallowtails. Near cuspidal cross caps, we also show that the sign of the Gaussian curvature is positive, that is, we can take only complex principal curvatures near cuspidal cross caps.