Long time existence for vortex filament equation in a Riemannian manifold (Q935566)

The author considers the dynamic of a curve \(\gamma(x,t)\) living in a Riemanian manifold \((M,g)\), described by the ``the vortex filament'' equation \[ \gamma_t=\gamma_x \times \nabla_x \gamma_x, \tag{1} \] where \(\times \) is the exterior product, \(\nabla_x\) is the covariant derivative in \(M\) and \(\| \gamma_x \| =1\). He proves that if \((M,g)\) is a 3-dimensional oriented and complete Riemanian manifold with bounded sectional curvature, equation ({1}) has a unique global solution, for any \(C^{\infty}\) closed initial curve \(\gamma_0(x)\), such that \(\| \nabla_x\gamma_x \| =1\).