On the regularity of timelike extremal surfaces

2012/11/13 by Robert L. Jerrard, Jerrard, Robert L., Matteo Novaga +3
Mathematics · #35L70 #53C44 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35L70 #msc:53C44

paper · pdf · doi:10.48550/arxiv.1211.3162

16 pages

arxiv created 2012/11/13 · arxiv updated 2012/11/15

Abstract

We study a class of timelike weakly extremal surfaces in flat Minkowski space \mathbb R¹⁺ⁿ, characterized by the fact that they admit a C¹ 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 R¹⁺ⁿ is 1-dimensional if n=2, and it is empty if n ≥ 4. For n=3, timelike weakly extremal surfaces exhibit an intermediate behavior.

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