Blow-up for the 3-dimensional axially symmetric harmonic map flow into S²

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S², ut = ∆u + |∇u|²u in Ω × (0, T) u = ub on ∂Ω × (0, T) u(·, 0) = u₀ in Ω, with u(x, t): Ω^¯ × [0, T) → S². Here Ω is a bounded, smooth axially symmetric domain in R³. We prove that for any circle Γ ⊂ Ω with the same axial symmetry, and any sufficiently small T > 0 there exist initial and boundary conditions such that u(x, t) blows-up exactly at time T and precisely on the curve Γ, in fact |∇u(·, t)|² * |∇u∗|² + 8πδ_Γ as t → T. for a regular function u∗(x), where δ_Γ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a spacecodimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5, 6].