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.
Funding Statement
The first author was supported by Grant-in-Aid for Scientific Research (17K05221), Japan Society for the Promotion of Science. This work was partly supported by Osaka City University Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.
Citation
Naoya ANDO. Kohei HAMADA. Kaname HASHIMOTO. Shin KATO. "Regularity of ends of zero mean curvature surfaces in $\mathbf{R}^{2,1}$." J. Math. Soc. Japan 74 (4) 1295 - 1334, October, 2022. https://doi.org/10.2969/jmsj/85018501
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