*Published Paper*

**Inserted:** 14 nov 2012

**Last Updated:** 13 sep 2015

**Journal:** Comm. Contemp. Math.

**Volume:** 17

**Number:** 1

**Pages:** 19 pages

**Year:** 2015

**Abstract:**

We study a class of timelike weakly extremal surfaces in at 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 is $1$-dimensional if $n = 2$, and it is empty if $n\geq 4$. For $n = 3$, timelike weakly extremal surfaces exhibit an intermediate behavior.

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