Embeddedness of Timelike Maximal Surfaces in $$(1+2)$$ ( 1 + 2 ) -Minkowski Space
Annales Henri Poincaré(2020)
Abstract
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.
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