Embeddedness of Timelike Maximal Surfaces in $$(1+2)$$ ( 1 + 2 ) -Minkowski Space

We prove that if $$\phi :\mathbb {R}^2 \rightarrow \mathbb {R}^{1+2}$$ is a smooth, proper, timelike immersion with vanishing mean curvature, then necessarily $$\phi $$ is an embedding, and every compact subset of $$\phi (\mathbb {R}^2)$$ 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 will be $$C^2$$ inextendible beyond some 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.