On the regularity of timelike extremal surfaces

TL;DR

This paper investigates the regularity and singularity structure of timelike extremal surfaces in Minkowski space, establishing bounds on singular set dimensions based on parametrization smoothness and analyzing generic singularity behavior.

Contribution

It provides sharp bounds on the Hausdorff dimension of singular sets for timelike extremal surfaces depending on parametrization regularity and characterizes generic singularity structures in various dimensions.

Findings

Singular set dimension is at most 1+1/k for $C^k$ parametrizations.

Generic timelike extremal cylinders have 1-dimensional singular sets in 3D.

Singular sets are empty for $n \,\geq\, 4$.

Abstract

We study a class of timelike weakly extremal surfaces in flat Minkowski space $\mathbb R^{1+n}$, characterized by the fact that they admit a $C^1$ parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class $C^k$, then the surface is regularly immersed away from a closed singular set of euclidean Hausdorff dimension at most $1+1/k$, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in $\mathbb R^{1+n}$ is 1-dimensional if $n=2$, and it is empty if $n \ge 4$. For $n=3$, timelike weakly extremal surfaces exhibit an intermediate behavior.