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

## TL;DR

This paper proves that smooth proper timelike maximal surfaces in (1+2) Minkowski space are embedded and graphical, and shows that evolutions of certain spacelike curves lead to singularities with curvature blow-up, preventing smooth continuation.

## Contribution

It establishes embeddedness and graphical nature of timelike maximal surfaces and analyzes singularity formation in their evolution from spacelike curves.

## Key findings

- Proper timelike maximal surfaces are embeddings and graphs.
- Evolving spacelike curves into timelike surfaces leads to singularities.
- Curvature blow-up prevents smooth extension beyond singularities.

## Abstract

We prove that if $\phi \colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$ 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 must undergo a scalar curvature blow-up---where the blow-up is with respect to an $L^1L^\infty$ 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.