We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+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 R^(1+n) is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.
On the regularity of timelike extremal surfaces
NOVAGA, MATTEO;
2015-01-01
Abstract
We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+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 R^(1+n) is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
