分野別セミナー

A timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is a surface with a Lorentzian metric whose mean curvature vanishes identically. One of the most important differences between spacelike surfaces (or surfaces in the Euclidean space) and timelike surfaces is the diagonalizability of the shape operator of asurface. For the minimal case, the diagonalizability of the shape operator corresponds to the sign of the Gaussian curvature of a timelike minimal surface away from flat points.

In this talk we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined only by the orientations of the two null curves that generate the surface. Moreover, we also discuss the behavior of the Gaussian curvature of a timelike minimal surface with some kind of singularities.