# On the regularity of timelike extremal surfaces

created by novaga on 14 Nov 2012
modified on 13 Sep 2015

[BibTeX]

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.