Regularity of ends of zero mean curvature surfaces in $\mathbf{R}^{2,1}$

Abstract

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