Long time existence for vortex filament equation in a Riemannian manifold

Abstract

Vortex filament equation in the Euclidean space has a long time solution for any closed initial data, because it can be converted into a standard nonlinear Schrödinger equation. While the Riemannian version of vortex filament equation is not integrable at all, we prove that it has a long time solution for any closed initial data.