Embeddedness of timelike maximal surfaces in (1+2) Minkowski space

We prove that if $\varphi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^2\to {\mathbb{R}}^{1+2}$ is a smooth proper timelike immersion with vanishing mean curvature, then necessarily $\varphi$ is an embedding, and every compact subset of $\varphi ({\mathbb{R}}^2)$ is a smooth graph. It follows that if one evolves any smooth self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in ${\mathbb{R}}^{1+2}$, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allowing for null points, such a Cauchy evolution must undergo a scalar curvature blow-up—where the blow-up is with respect to an $L^1{L}^{\mathrm{\infty }}$ norm—and thus the evolving surface will be $C^2$ inextendible beyond singular time. In addition we study the continuity of the unit tangent for the evolution of a self-intersecting curve in isothermal gauge, which defines a well-known evolution beyond singular time.