On the regularity of timelike extremal surfaces

Abstract

We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+n, characterized by the fact that they admit a C1 parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class Ck, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1 + 1/k, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in ℝ1+n is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.