Regularity of ends of zero mean curvature surfaces in 𝐑^2,1

Abstract

In this paper, we analyze ends of zero mean curvature surfaces of mixed (or non-mixed) type in the Lorentzian 3-space 𝐑^2,1. Among these, we show that spacelike or timelike planar ends are 𝐶^∞ in the compactification \hat{𝐿} of 𝐑^2,1 as in the case of minimal surfaces in the Euclidean 3-space 𝐑³. On the other hand, lightlike planar ends are not 𝐶^∞. Each lightlike planar end of a mixed type surface has the following additional parts: the converging part (a lightlike line in 𝐑^2,1), the diverging part (the set of the points in \hat{𝐿} ∖ 𝐑^2,1 corresponding to zero-divisors), and the border of these two parts. We show that such an end is 𝐶^∞ on the first two parts almost everywhere, while there appears an isolated singularity in the form of (𝑥³, 𝑥𝜏 + “higher order terms”, 𝜏) on the border. We also show that conelike singularities of mixed type appear on the lightlike lines in special cases.