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We prove that if $\phi \colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$ is a smoothproper timelike immersion with vanishing mean curvature, then necessarily$\phi$ is an embedding, and every compact subset of $\phi(\mathbb{R}^2)$ is asmooth graph. It follows that if one evolves any smooth self-intersectingspacelike curve (or any planar spacelike curve whose unit tangent vector spansa closed semi-circle) so as to trace a timelike surface of vanishing meancurvature in $\mathbb{R}^{1+2}$, then the evolving surface will either fail toremain timelike, or it will fail to remain smooth. We show that, even allowingfor null points, such a Cauchy evolution must undergo a scalar curvatureblow-up---where the blow-up is with respect to an $L^1L^\infty$ norm---and thusthe evolving surface will be $C^2$ inextendible beyond singular time. Inaddition we study the continuity of the unit tangent for the evolution of aself-intersecting curve in isothermal gauge, which defines a well-knownevolution beyond singular time.

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