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

Abstract

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