We consider the Ginzburg-Landau energy E ε for R M -valued maps defined in a cylinder shape domain B N × (0, 1) n satisfying a degree-one vortex boundary condition on ∂B N × (0, 1) n in dimensions M ≥ N ≥ 2 and n ≥ 1. The aim is to study the radial symmetry of global minimizers of this variational problem. We prove the following: if N ≥ 7, then for every ε > 0, there exists a unique global minimizer which is given by the non-escaping radially symmetric vortex sheet solution

and M ≥ N + 1, the following dichotomy occurs between escaping and non-escaping solutions: there exists ε N > 0 such that • if ε ∈ (0, ε N ), then every global minimizer is an escaping radially symmetric vortex sheet solution of the form Rũ ε where ũε (x, z) = ( fε (|x|

• if ε ≥ ε N , then the non-escaping radially symmetric vortex sheet solution

) n is the unique global minimizer; moreover, there are no bounded escaping solutions in this case.

We also discuss the problem of vortex sheet S M-1 -valued harmonic maps.